Mathematical Modelling of Drug Delivery to Solid Tumour · Solid Tumour by Wenbo ZHAN A ......

Mathematical Modelling of Drug Delivery to

Solid Tumour

by

Wenbo ZHAN

A Thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy and Diploma of Imperial College London

Department of Chemical Engineering

Imperial College London

September 2014

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Abstract

Effective delivery of therapeutic agents to tumour cells is essential to the success of most cancer

treatment therapies except for surgery. The transport of drug in solid tumour involves multiple

biophysical and biochemical processes which are strongly dependent on the physicochemical

properties of the drug and biological properties of the tumour. Owing to the complexities

involved, mathematical models are playing an increasingly important role in identifying the

factors leading to inadequate drug delivery to tumours. In this study, a computational model is

developed which incorporates real tumour geometry reconstructed from magnetic resonance

images, drug transport through the tumour vasculature and interstitial space, as well as drug

uptake by tumour cells. The effectiveness of anticancer therapy is evaluated based on the

percentage of survival tumour cells by directly solving the corresponding pharmacodynamics

equation using predicted intracellular drug concentration.

Computational simulations have been performed for the delivery of doxorubicin through various

delivery modes, including bolus injection and continuous infusion of doxorubicin in free form,

and thermo-sensitive liposome mediated doxorubicin delivery activated by high intensity focused

ultrasound. Predicted results show that continuous infusion is far more effective than bolus

injection in maintaining high levels of intracellular drug concentration, thereby increasing drug

uptake by tumour cells. Moreover, multiple-administration is found to be more effective in

improving the cytotoxic effect of drug compared to a single administration.

The effect of heterogeneous distribution of microvasculature on drug transport in a realistic

model of liver tumour is investigated, and the results indicate that although tumour interstitial

fluid pressure is almost uniform, drug concentration is sensitive to the heterogeneous distribution

of microvasculature within a tumour. Results from three prostate tumours of different sizes

suggest a nonlinear relationship between transvascular transport of anticancer drugs and tumour

size.

Numerical simulations of thermo-sensitive liposome-mediated drug delivery coupled with high

intensity focused ultrasound heating demonstrate the potential advantage of this novel drug

delivery system for localised treatment while minimising drug concentration in normal tissue.

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Acknowledgement

This thesis represents not only the result of my hard work over the last four years, but also

invaluable supports from sincere and kind people in my life.

First and foremost, I am grateful to my supervisor Professor Xiao Yun Xu. She has been

professionally and personally supportive since the day I applied for a Ph.D. position at Imperial

College. My Ph.D. experience would not have been as stimulating and productive without her

contributions of ideas and time. I appreciate her motivational enthusiasm and great patience to

guide me through the tough time in the Ph.D. pursuit.

Many people have helped and taught me immensely during this project. Thank you to Professor

Wladyslaw Gedroyc for kindly providing MR images and answering all my questions. I would

like to acknowledge Professor Simon Thom who gave me many suggestions from clinical

aspects to help improve my work. The time of working with Professor Nigel Wood is

unforgettable because of his great enthusiasm and appreciable willingness. I also want to express

my special thanks to Dr. Xin Zhang for sharing knowledge of thermo-sensitive liposome and

providing useful suggestions based on his experiments. Generous help was also given by Dr.

Cong Liu, who helped me a lot in deepening understanding of drug transport processes.

Our group has become a source of friendship, kind help and collaboration. My time at the college

could not have been as colourful or joyful without any of these. Thanks to Drs. Ryo Torii, Zhou

Cheng and Chrysa Anna Kousera, my fellow colleagues Afet Mehmet, Zhongjie Wang,

Harkamaljot Kandail, Shelly Singh, Oluwatoyin Fatona, Ana Crispin Corzo, Boram Gu, Claudia

Menichini and Andris Piebalgs. I wish you all the best.

My time in the U.K. was made enjoyable and cheerful because of many friends. I cherish the

time spent with friends in London, trips to other UK cities and abroad. Many thanks to my

friends for their long-lasting friendship over years. Da Gao and Wenbo Dong’s visits were

surprising gifts to me. Heartfelt thanks to Yijiu Jiang for his fraternal supports in the past years.

Last but not the least, I am deeply indebted to my parents for their love and encouragement under

any circumstances. They raised me with a love of science and supported me in all reasonable

pursuits. It is my great honour and happiness to be a member of the family. Thank you.

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Declaration

I hereby declare that the work represented in this thesis was originated entirely from me.

Information derived from the published and unpublished work of others has been acknowledged

in the text and references are given in the bibliography.

The copyright of this thesis rests with the author and is made available under a Creative

Commons Attribution-Non Commercial-No Derivatives licence. Researchers are free to copy,

distribute or transmit the thesis on the condition that they attribute it, that they do not use it for

commercial purposes and that they do not alter, transform or build upon it. For any reuse or

distribution, researchers must make clear to others the licence terms of this work.

Wenbo ZHAN

Imperial College London, September 2014

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Contents

1. Introduction ........................................................................................................................... 17

1.1. Background .................................................................................................................... 18

1.1.1. Cancer ..................................................................................................................... 18

1.1.2. Anticancer Therapy ................................................................................................. 19

1.1.3. Thermo-Sensitive Liposome-Mediated Drug Delivery .......................................... 22

1.2. Objectives of the Study .................................................................................................. 23

1.3. Structure of This Thesis ................................................................................................. 24

2. Literature Review ................................................................................................................. 25

2.1. Characteristics of Solid Tumour .................................................................................... 26

2.1.1. Tumour Vasculature................................................................................................ 26

2.1.2. Blood Flow Rate ..................................................................................................... 28

2.1.3. Lymphatic Drainage................................................................................................ 29

2.1.4. Interstitial Fluid Pressure ........................................................................................ 30

2.2. Modes of Drug Delivery to Solid Tumour ..................................................................... 30

2.2.1. Traditional Administration...................................................................................... 31

2.2.2. Nanoparticle-Mediated Delivery ............................................................................ 32

2.2.3. Polymer Implant...................................................................................................... 34

2.3. Drug Transport in Solid Tumour .................................................................................... 34

2.3.1. Transport within Blood Vessel ............................................................................... 35

2.3.2. Transport through Blood Vessels............................................................................ 36

2.3.3. Transport in Interstitial Space ................................................................................. 37

2.3.4. Transport to Lymphatic System .............................................................................. 39

2.3.5. Transport through Cell Membrane.......................................................................... 39

2.3.6. Cell Killing.............................................................................................................. 40

2.4. Summary ........................................................................................................................ 41

3. Mathematical Models and Numerical Procedures............................................................. 43

3.1. Mathematical Models ..................................................................................................... 44

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3.1.1. Interstitial Fluid Flow ............................................................................................. 45

3.1.2. Drug Transport ........................................................................................................ 47

3.1.3. Thermo-Sensitive Liposome-Mediated Drug Transport ......................................... 49

3.1.4. Pharmacodynamics Model ...................................................................................... 51

3.1.5. Heat Transfer under High Intensity Focused Ultrasound (HIFU) Heating ............ 51

3.2. Model Parameters ........................................................................................................... 53

3.2.1. Tissue Related Transport Parameters ...................................................................... 54

3.2.2. Drug Related Transport Parameters ........................................................................ 54

3.3. Implementations of Mathematical Models ........................................................................... 59

3.3.1. ANSYS Fluent ........................................................................................................ 59

3.3.2. User Defined Routines for Drug Transport ............................................................ 60

3.3.3. User Defined Routines for Heat Transfer ............................................................... 60

3.3.4. Boundary Conditions .............................................................................................. 60

3.4. Summary ........................................................................................................................ 60

4. Non-Encapsulated Drug Delivery to Solid Tumour .......................................................... 65

4.1. Model Description .......................................................................................................... 66

4.1.1. Model Geometry ..................................................................................................... 66

4.1.2. Model Parameters ................................................................................................... 67

4.1.3. Numerical Methods ................................................................................................. 69

4.2. Results ............................................................................................................................ 70

4.2.1. Interstitial Fluid Field ............................................................................................. 70

4.2.2. Single Administration Mode ................................................................................... 71

4.2.2.1. Infusion Duration .................................................................................................. 71

4.2.2.2. Effect of Dose Level ............................................................................................. 76

4.2.3. Multiple Administration Mode ............................................................................... 78

4.2.3.1. Continuous Infusion .............................................................................................. 79

4.2.3.2. Bolus Injection ...................................................................................................... 83

4.2.4. Comparison of 2-D and 3-D Models ...................................................................... 86

4.3. Discussion ...................................................................................................................... 87

4.4. Summary ........................................................................................................................ 89

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5. Effect of Tumour Properties on Drug Delivery ................................................................. 90

5.1. Heterogeneous Vasculature Distribution ....................................................................... 91

5.1.1. Model Geometry ..................................................................................................... 91

5.1.2. Model Parameters ................................................................................................... 92

5.1.3. Numerical Methods ................................................................................................. 93

5.1.4. Results ..................................................................................................................... 94

5.1.4.1. Interstitial Fluid Field ........................................................................................... 94

5.1.4.2. Doxorubicin Concentration ................................................................................... 95

5.1.4.3. Doxorubicin Cytotoxic Effect ............................................................................... 97

5.1.4.4. Comparison of Various Administration Schedules ............................................... 98

5.2. Tumour Size ................................................................................................................... 99

5.2.1. Model Geometry ..................................................................................................... 99

5.2.2. Model Parameters ................................................................................................. 100

5.2.3. Results ................................................................................................................... 101

5.2.3.1. Interstitial Fluid Field ......................................................................................... 101

5.2.3.2. Doxorubicin Concentration ................................................................................. 103

5.2.3.3. Doxorubicin Cytotoxic Effect ............................................................................. 107

5.3. Discussion .................................................................................................................... 108

5.4. Summary ...................................................................................................................... 110

6. Thermo-Sensitive Liposome-Mediated Drug Delivery to Solid Tumour....................... 111

6.1. Acoustic Model for High Intensity Focused Ultrasound ............................................. 112

6.2. Thermo-Sensitive Liposome-Mediated Delivery ......................................................... 113

6.2.1. Model Description ................................................................................................ 113

6.2.1.1. Model Geometry ................................................................................................. 113

6.2.1.2. Model Parameters ............................................................................................... 114

6.2.1.3. Numerical Methods ............................................................................................. 119

6.2.2. Results and Discussion ......................................................................................... 120

6.2.2.1. Temperature Profile ............................................................................................ 121

6.2.2.2. Drug Concentration ............................................................................................. 123

6.2.2.3. Doxorubicin Cytotoxic Effect ............................................................................. 130

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6.3. Comparison of 2-D and 3-D Models ............................................................................ 132

6.4. Summary ...................................................................................................................... 135

7. Conclusion ........................................................................................................................... 136

7.1. Main Conclusion .......................................................................................................... 137

7.2. Limitations ................................................................................................................... 138

7.2.1. Tumour Property ................................................................................................... 139

7.2.2. Drug Transport Model .......................................................................................... 139

7.2.3. HIFU Heating Model ............................................................................................ 140

7.3. Suggestions for Future Work ....................................................................................... 140

7.3.1. Realistic Blood Vessel Geometry ......................................................................... 140

7.3.2. Nutrient Transport ................................................................................................. 141

7.3.3. Model Validation .................................................................................................. 141

7.3.4. Type / Patient-Specific Tumour ............................................................................ 141

7.3.5. Chemotherapy Drugs ............................................................................................ 142

7.3.6. HIFU Heating Strategy ......................................................................................... 142

Appendix: Publications during the Project ........................................................................... 143

Appendix: Nomenclature ........................................................................................................ 144

Bibliography .............................................................................................................................. 149

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List of Figures

Figure 1-1. World cancer death 2012 ........................................................................................... 18

Figure 2-1. Normal and tumor vasculature. .................................................................................. 26

Figure 2-2. Typical architecture of blood vessels in solid tumour ............................................... 27

Figure 2-3. Regions in Solid Tumour ........................................................................................... 28

Figure 2-4. Schematic diagram of drug transport ......................................................................... 34

Figure 2-5. Schematic diagram of three-compartment model ...................................................... 35

Figure 3-1. Drug transport with (a) continuous infusion of drug in its free form, and (b) thermo-

sensitive liposome-mediated drug delivery ................................................................ 45

Figure 3-2. Schematic representation of heat transfer under HIFU heating in (a) tumour and (b)

blood........................................................................................................................... 51

Figure 3-3. Dimensions and coordinates of a spherical radiator .................................................. 53

Figure 3-4. Estimation of doxorubicin diffusion coefficient in tumour tissue.. ........................... 56

Figure 4-1. Model geometry: (a) MR image of the prostate tumour (in red) and its surrounding

tissue (in pule blue); (b) the reconstructed 2-D geometry. ........................................ 67

Figure 4-2. Numerical Procedures. ............................................................................................... 69

Figure 4- 3. Interstitial fluid pressure in tumour and surrounding normal tissue. ........................ 70

Figure 4-4. Doxorubicin concentration in plasma as a function of time after start of treatment, for

bolus injection and continuous infusion of various indicated durations (dose = 50

mg/m2). ....................................................................................................................... 72

Figure 4-5. Spatial mean (a) free and (b) bound doxorubicin extracellular concentration in

tumour as a function of time under bolus injection and different infusion durations

(dose = 50 mg/m2). ..................................................................................................... 73

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Figure 4-6. Spatial mean (a) free and (b) bound doxorubicin extracellular concentration in

normal tissue as a function of time under bolus injection and different infusion

durations. (dose = 50 mg/m2). .................................................................................... 73

Figure 4-7. Spatial distribution of extracellular concentration of free doxorubicin in tumour and

normal tissues (2-hour infusion, time = 0.5 hour). .................................................... 74

Figure 4-8. Doxorubicin intracellular concentration as a function of time, for bolus injection and

continuous infusions of various durations (dose = 50 mg/m2). ................................. 74

Figure 4-9. Predicated percentage tumour cell survival under bolus injection and continuous

infusions of various durations (dose = 50 mg/m2). .................................................... 75

Figure 4-10. Spatial mean doxorubicin concentration in tumour as a function of time under 2-

hour infusion duration.(a) doxorubicin blood concentration, (b) free doxorubicin

extracellular concentration, (c) bound doxorubicin extracellular concentration, and (d)

intracellular concentration. ......................................................................................... 77

Figure 4-11. Predicated percentage of tumour cell survival under various doses. ....................... 78

Figure 4-12. Spatial mean doxorubicin concentration in normal tissue as a function of time under

2-hour infusion duration (for a 70 kg patient). (a) free doxorubicin extracellular

concentration, (b) bound doxorubicin extracellular concentration. ........................... 78

Figure 4-13. Doxorubicin intravascular concentration under single and multiple continuous

infusions as a function of time (total dose = 50 mg/m2). ........................................... 79

Figure 4-14. Spatial mean doxorubicin extracellular concentration as a function of time under

single and multiple continuous infusion. (a) free doxorubicin, and (b) bound

doxorubicin (total dose = 50 mg/m2). ........................................................................ 80

Figure 4-15. Time course of doxorubicin intracellular concentration under single and multiple

continuous infusions. (total dose = 50 mg/m2) .......................................................... 80

Figure 4-16. Time course of tumour survival fraction under single and multiple continuous

infusions. .................................................................................................................... 81

Figure 4-17. Spatial mean doxorubicin extracellular concentration in normal tissue as a function

of time under single and multiple continuous infusion against time. (a) free

doxorubicin, (b) bound doxorubicin (total dose = 50 mg/m2). .................................. 82

Figure 4-18. Comparison of various doses under single and multiple-infusions. (a) intracellular

concentration and (b) cell survival fraction under single infusion with 50 mg/m2 and

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multiple infusions with 25 mg/m2 as a function of time ; (c) intracellular

concentration and (d) cell survival fraction under single infusion with 75 mg/m2 and

multiple infusions with 50 mg/m2 as a function of time. ........................................... 83

Figure 4-19. Comparison of doxorubicin concentration under single and multiple bolus injections

as a function of time (total dose = 50 mg/m2). (a) intravascular concentration, (b) free

doxorubicin extracellular concentration, (c) bound doxorubicin extracellular

concentration, and (d) intracellular concentration ..................................................... 84

Figure 4-20. Tumour survival fraction as a function of time with single and multiple bolus

injection ...................................................................................................................... 85

Figure 4-21. Comparison of doxorubicin concentration under single and multiple bolus injections

as a function of time (total dose = 50 mg/m2). (a) free and (b) bound doxorubicin

concentration in normal tissue. .................................................................................. 85

Figure 4-22. Comparison of survival fraction between single continuous infusion and 8 times

bolus injection. ........................................................................................................... 86

Figure 4-23. 3-D model geometry. ............................................................................................... 86

Figure 4-24. Comparison of drug concentration and cell density between 2-D and 3-D models. (a)

free doxorubicin extracellular concentration in tumour as a function of time, (b)

maximum percentage differences in drug concentration and tumour cell density

(total dose = 50 mg/m2) .............................................................................................. 87

Figure 5-1. Model geometry: (a) MR image of the liver tumour (in red) and its surrounding tissue

(in pule blue); (b) the reconstructed 2-D geometry. ................................................... 92

Figure 5-2. Interstitial fluid pressure distribution in tumour and normal tissues.......................... 95

Figure 5-3. Spatial distribution of free doxorubicin extracellular concentration in tumour regions

(total dose = 50 mg/m2, infusion duration = 2 hours) ................................................ 95

Figure 5-4. Spatial mean doxorubicin extracellular concentration in tumour as a function of time.

(a) Free doxorubicin, and (b) bound doxorubicin. (total dose = 50 mg/m2, infusion

duration = 2 hours) ..................................................................................................... 96

Figure 5-5. Spatial mean of doxorubicin intracellular concentration in tumour regions as a

function of time. (total dose = 50 mg/m2, infusion duration = 2 hours) .................... 96

13

Figure 5-6. Spatial mean percentage of tumour cells in tumour regions as a function of time (total

dose = 50 mg/m2, infusion duration = 2 hours). ........................................................ 97

Figure 5-7. Spatial distribution of survival cell fraction in tumour regions (total dose = 50 mg/m2,

infusion duration = 2 hours) ....................................................................................... 98

Figure 5-8. Maximum difference among tumour regions over treatment with various doses. (a)

free doxorubicin extracellular concentration and (b) intracellular concentration

(infusion duration = 2 hours) ..................................................................................... 98

Figure 5-9. Maximum difference among tumour regions over treatment with various infusion

durations. (a) free doxorubicin extracellular concentration and (b) intracellular

concentration (total dose = 50 mg/m2) ....................................................................... 99

Figure 5-10. Model geometry: (a) case_1; (b) case_2; (c) case_3 .............................................. 100

Figure 5-11. Estimation of blood vessel surface area to tissue volume ratio of tumours with

various tumour sizes................................................................................................. 100

Figure 5-12. Interstitial fluid pressure distribution in tumour and normal tissue ....................... 102

Figure 5-13. The spatial mean interstitial fluid pressure and transvacular flux per tumour volume

as a function of tumour size ..................................................................................... 102

Figure 5-14. Free and bound doxorubicin concentration in blood after administration as a

function of time. (infusion duration = 2 hours, total dose = 50mg/m2) ................... 103

Figure 5-15. Spatial mean doxorubicin extracellular concentration as a function of time under 2-

hour continuous infusion, dose = 50mg/m2. (a) Free and (b) bound doxorubicin in

tumour, (c) bound and (d) bound doxorubicin in normal tissue .............................. 104

Figure 5-16. Doxorubicin exchange between blood vasculature and interstitial fluid by

convection as a function of time. (a) free and (b) doxorubicin exchange (2-hour

infusion, total dose =50 mg/m2) ............................................................................... 105

Figure 5-17. Doxorubicin exchange between blood vasculature and interstitial fluid as a function

of time in each tumour. (a) free and (b) bound doxorubicin exchange by diffusion

owing to concentration gradient (2-hour infusion, total dose = 50 mg/m2) ............. 105

Figure 5-18. Comparison of (a) free and (b) bound doxorubicin exchange between blood and

interstitial fluid by convection and diffusion in case_2 (2-hour infusion, total dose =

50mg/m2) .................................................................................................................. 107

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Figure 5-19. Temporal profiles of predicated intracellular concentration in tumours with different

sizes .......................................................................................................................... 107

Figure 5-20. Survival cell fraction of tumour as a function of time in three tumours with different

sizes (2-hour infusion, total dose = 50 mg/m2) ........................................................ 108

Figure 6-1. Spatial distribution of acoustic pressure .................................................................. 112

Figure 6-2. Measured and computed acoustic pressure profiles at 1.33 MHz and 1.45 MPa

pressure at the focal point. (a) along axis and (b) radial distance.. .......................... 112

Figure 6-3. Model geometry and locations of focus regions. ..................................................... 113

Figure 6-4. Permeability of free doxorubicin as a function of temperature... ............................. 116

Figure 6-5. Fold increase in permeability of liposome-mediated doxorubicin as a function of

temperature............................................................................................................... 116

Figure 6-6. Viscosity of water as a function of temperature....................................................... 117

Figure 6-7. Drug release at (a) 37oC and (b) 42

oC. ..................................................................... 119

Figure 6-8. Numerical Procedure ................................................................................................ 120

Figure 6-9. Variations of temperature in tumour, blood and normal tissue as a function of time

for 10 minutes heating. (a) Maximum and (b) spatial-mean temperature. .............. 121

Figure 6-10. Spatial distribution of temperature in (a) tissue and (b) blood at 10min after heating

starts. ........................................................................................................................ 122

Figure 6-11. Variations of temperature in tumour, blood and normal tissue as a function of time

for various heating durations. (a) Maximum and (b) spatial-mean temperature for 30

minutes heating; (c) Maximum and (d) spatial-mean temperature for 60 minutes

heating. ..................................................................................................................... 123

Figure 6-12. Spatial-mean concentration of encapsulated doxorubicin in blood stream as a

function of time (dose = 50 mg/m2). (a) in tumour, and (b) in normal tissue. ......... 124

Figure 6-13. Encapsulated doxorubicin concentration in blood at 10 min after heating ............ 124

Figure 6-14. Spatial-mean concentration (a) and flux (b) of encapsulated doxorubicin in tumour

extracellular space as a function of time. ................................................................. 125

Figure 6-15. Spatial-mean concentration of encapsulated doxorubicin in extracellular space of

normal tissue as a function of time. ......................................................................... 126

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Figure 6-16. Spatial distribution of encapsulated doxorubicin extracellular concentration in

tumour and surrounding normal tissues at various time points with 60 minutes

heating. ..................................................................................................................... 126

Figure 6-17. Doxorubicin extracellular concentration with various heating duration as a function

of time. (a) free and (b) doxorubicin concentration in tumour, and (c) free and (d)

doxorubicin concentration in normal tissue. ............................................................ 127

Figure 6-18. Doxorubicin extracellular concentration with various heating duration as a function

of time. (a) free and (b) doxorubicin concentration in tumour, and (c) free and (d)

doxorubicin concentration in normal tissue ............................................................. 128

Figure 6-19. Spatial distribution of doxorubicin concentration and cell survival fraction in

tumour at 10 minutes................................................................................................ 129

Figure 6-20. Doxorubicin intracellular concentration with heating and no-heating as a function of

time........................................................................................................................... 130

Figure 6-21. Survival cell fraction in treatment with various heating duration as a function of

time........................................................................................................................... 131

Figure 6-22. Improvement by HIFU heating. (a) intracellular concentration, and (b) cell killing

.................................................................................................................................. 131

Figure 6-23. Comparison of maximum temperature of tumour in 2-D and 3-D model. ............ 133

Figure 6-24. Comparison of spatial mean temperature in 2-D and 3-D model. (a) tumour and (b)

normal temperature. ................................................................................................. 133

Figure 6-25. Comparison of spatial mean temperature in 2-D and 3-D model. (a) blood

temperature in tumour, and (b) blood temperature in normal tissue. ....................... 134

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List of Tables

Table 1-1. Processes affected by chemotherapeutic drugs ........................................................... 20

Table 1-2. Common side effects caused by chemotherapeutic drugs ........................................... 20

Table 2-1. Morphological features of tumour microvasculature .................................................. 27

Table 2-2. Mean blood flow rate in human tumour and normal tissue ......................................... 29

Table 3-1. Parameters for tumour and normal tissues. ................................................................. 62

Table 3-2. Parameters for liposome .............................................................................................. 63

Table 3-3. Parameters for doxorubicin. ........................................................................................ 64

Table 4-1. MR imaging parameters .............................................................................................. 66


Table 5-2. The relative and scaled values of surface area of blood vessel per tumour size ......... 93


Table 5-4. Tumour size and the blood vessel surface area to tissue volume ratio in each model

..................................................................................................................................................... 101

Table 6-1. Acoustic and thermal properties ................................................................................ 114

Table 6-2. HIFU transducer parameters...................................................................................... 114

Table 6-3. Release rates at various temperatures ........................................................................ 119

1. Introduction

Cancer is a major cause of mortality and morbidity in the developed and developing

countries. Currently, cancer is treated primarily by surgery, radiation therapy and

chemotherapy. Novel approaches, such as gene therapy, immunotherapy and

antiangiogenesis therapy, have been developed extensively and are likely to become

clinically available in the next few decades. Except for surgery, effective delivery of

therapeutic agents to tumour cells is the key to success for these therapies. Therefore,

in this thesis, the essential physical and biochemical processes involved in drug

delivery to solid tumours are studied. This chapter presents the relevant background

and preliminaries of the project. The objectives and strategies of the research are then

defined, which is followed by an outline of the thesis.

1. INTRODUCTION

18

1.1. Background

1.1.1. Cancer

Cancer is a set of diseases characterized by unregulated cell growth leading to invasion of

surrounding tissues and spread to other parts of the body [1]. Cancer cells can divide and grow

uncontrollably, forming malignant tumours and finally invading nearby normal tissues. Up to

now, over 200 types of cancer have been discovered. As shown in Figure 1-1, the high mortality

rate associated with cancer makes it one of the most serious life-threatening diseases in the world.

Around 159,000 people died from cancer in the UK in 2011, indicating 252 deaths for every

100,000 people.

116.0+

99.6 – 116.0

89.7 – 99.6

73.3 – 89.7

<73.3

No Data

Death per 100,000

Population*

*All cancers except non-melanoma skin cancer

Figure 1-1. World cancer death 2012

The causes of cancer are diverse, complex and not totally understood [2]. Various factors have

been suggested to increase the risk of cancer, including pollution, radiation, industrial products

and certain types of infections caused by damaged genes or combined with existing genetic faults

within cells. Cancerous mutations of proliferation related genes, proto-oncogenes and/or tumour

suppressor genes, drive cancer cells to go on duplicating themselves.

Cancer seems to develop progressively [2]. As cancer cells expand, the original space becomes

insufficient for their uncontrolled growth, resulting in invasion of increasing number of cells to

the adjacent normal tissues. In order to acquire oxygen and nutrition to support cancer growth,

normal cells are signalled to help develop new blood vessels. These vessels tend to be leaky and

with abnormal geometric properties, which are different from blood vessels in normal tissues. In

1. INTRODUCTION

19

addition, cancer cells can migrate from the primary cancer and spread to more distant parts of the

body through the blood stream or lymphatic system to establish new colonies of cancer cells.

These new settlements, termed metastases, are the major cause of death since vital organs can be

damaged by cancer cells [1]. Ideally, treatments should start before cancer cells invade into other

organs.

1.1.2. Anticancer Therapy

Various anticancer treatments are available in clinics, and the choice of these treatments is

strongly dependent on the location, growth stage of cancer as well as the health condition of

patients, etc. The main anticancer therapies are outlined below.

(1) Surgery

Surgery is cutting away tissue from patient’s body. Given that it is a local treatment, surgery may

cure a localised cancer that is totally contained in one area before metastases taking place. In

surgery, tumour and some of its surrounding normal tissues are removed to make the margin

clear. Considering that cancer may invade the lymphatic system, the nearest lymph nodes are

also removed. This is a preferred treatment for cancer, and can offer a complete cure in some

cases [1].

After spreading of cancer cells, surgery usually cannot cure the cancer, but may also help

patients live longer and reduce/control pain. Another application of surgery is to prevent or

reduce the risk of cancer. People at a high risk of a particular type of cancer (e.g. bowel or breast

cancer) may choose surgery to remove the relevant tissue.

Surgery is limited by the type of cancer. Certain types of cancer are difficult to be removed due

to their location, and insufficient removal may result in regrowth of cancer. On the other hand,

spalling of cancer cells during surgery may cause the spread of cancer cells.

(2) Chemotherapy

Chemotherapy is the use of cytotoxic or cytostatic drugs that respectively kill malignant cells or

prevent them from proliferating. The effectiveness of chemotherapy strongly depends on the

1. INTRODUCTION

20

amount of drugs accumulated in the tumour, exposure duration to drugs and the proportion of

cell population that is proliferating.

There are more than 100 different anticancer drugs currently available and new drugs are being

developed all the time. The choice of drugs is highly related to the type and stage of tumour.

Chemotherapy kills tumour cells by damaging the genes that make cells divide, or it may

interrupt the chemical process involved in cell division. The process affected by normal

chemotherapy drugs are summarized in Table 1-1.

Table 1-1. Processes affected by chemotherapeutic drugs [1]

Drug Process Affected Cancer Treated

Cisplatin DNA synthesis Ovary, Testis

Methotrexate Dihydrofolate reductase Breast, Placenta

5-Fluorouracil Thymidylate synthase, RNA synthesis Stomach, Breast, Colorectal

Doxorubicin DNA & RNA synthesis Lung, Breast, Leukaemia

Vincristine Tubulin polymerisation Lymphoma, Testis

Chemotherapy is a systemic treatment. Drugs are circulated around the body in the bloodstream

and can reach cells almost anywhere. Without specifying the target, normal cells can also be

killed and hence the normal function of tissue can be compromised. This is known as side effects,

which can seriously offset the treatment outcome. Side effects vary according to the drug, but

there are some most common side effects, as shown in Table 1-2.

Table 1-2. Common side effects caused by chemotherapeutic drugs [1]

Drug Side Effects

Cisplatin Nephrotoxicity, Ototoxicity, etc

Methotrexate Myelosuppression, Alopecia, Dermatitis, etc

5-Fluorouracil Myelosuppression, Cardiac toxicity, etc

Doxorubicin Cardiomyopathy, Myelosuppression, Leukopenia, etc

Vincristine Chemotherapy-induced peripheral neuropathy, Hyponatremia, etc

During chemotherapy, drugs often stop the bone marrow from generating enough blood cells.

Without adequate white blood cells, risks of developing infections will increase, which low

levels of red blood cells result in the shortage of nutrition and oxygen supply, causing

1. INTRODUCTION

21

breathlessness and tiredness. Most drugs cause hair loss, but this effect is temporary and hair

starts to grow back a few weeks after chemotherapy ends.

Changes in the way that liver, kidney, lung and heart work can also be caused by chemotherapy

drugs. These changes are usually temporary, but can be permanent for some patients.

Doxorubicin is a typical anticancer drug, and its most serious side effect is cardiomyopathy,

which can lead to heart failure. In order to minimise the risk for this, the total dose of

doxorubicin in a life cycle is limited to 550 mg per unit body surface area [3].

(3) Radiotherapy

Radiation is applied to kill malignant tumour cells by causing irreversible damages to DNA

within cancer cells. Since tumour stem cells are not as sensitive as cycling cells to radiation, the

entire treatment process may last longer than other types of treatment. According to the way

radiation is used, basically there are two types of radiotherapy: external radiotherapy and internal

radiotherapy.

In external radiotherapy, a narrow radiation beam with high intensity is generated to a well

identified area of the patient’s body. X-ray and gamma ray are the most common radiation used

clinically. Since radiation can pass through the body, superficial and deep-seated tumours can be

treated without surgical trauma. Given that the treated area is well identified before radiotherapy,

damage to adjacent and overlying normal cells can be minimized by re-directing the radiation

beam to restrict the radiation dose only being focused in tumour.

In internal radiotherapy, radioactive implants are inserted within or close to the target tumour,

and a radioactive liquid is administrated orally or injected first, which is then transported by the

circulation system. The radioactive component of the liquid is called isotope, which is designed

to attach to a 'bioconjugate' molecule, such as an antibody, to cause the liquid to be taken up by

tumour cells.

Although radiotherapy is a local treatment, some normal cells are inevitably killed, resulting in

undesirable side effects. During the treatment, tiredness is a common side effect since healthy

cells need to be repaired or the level of blood cells drops. In addition, radiotherapy can also

cause side effects in particular organs, such as in the head, neck or chest.

1. INTRODUCTION

22

(4) Novel and Emerging Therapies

High intensity focused ultrasound (HIFU) is a novel and fast developing therapy [4]. High

frequency ultrasound waves are emitted from an ultrasound transducer, and the waves deliver a

strong beam to the targeted region in order to elevate the temperature to kill tumour cells. It is a

highly targeted and localised treatment. Because of the non-invasive nature of the sound wave

used, the associated side effect is less than the other treatment methods. Patients may feel

localised pain in the first 3~4 days.

Photodynamic therapy (PDT) is usually used to treat non-melanoma skin cancer [5]. Drugs

(known as photosensitiser) are firstly administrated to add a specific type of light sensitivity to

tumour cells, which are then killed by exposure to this light. However, PDT can make skin and

eyes sensitive to light, and may cause death of nearby normal cells.

Gene therapy involves inserting specific genes into malignant or normal cells in order to modify

their gene expressions. Either viral or non-viral agents can be used to deliver the specific genes,

and viral must be attenuated before used. This therapy is with high specificity, but is still in

development stage. A number of early stage clinical trials have been carried out [6-8].

Clinically, the treatment methods are strongly dependent on the type, location and growth stage

of tumour as well as the health condition of patient. Usually, various treatments are combined in

order to improve the overall treatment outcome. Most of these therapies depend on effective

delivery of therapeutic agents to the targeted tumour.

1.1.3. Thermo-Sensitive Liposome-Mediated Drug Delivery

For an effective delivery, adequate therapeutic agents should reach the tumour while drug

accumulation in normal tissues needs to be prevented or reduced in order to minimize side

effects. Since most therapeutic agents are carried by systemic bloodstream to the entire body,

their usages are hence limited. As an improvement, thermo-sensitive liposome-mediated drug

delivery has been developed to overcome this shortcoming.

Anticancer drugs are encapsulated by thermo-sensitive liposomal particles which are

administrated into the blood stream as usual [9]. Ideally, these particles should retain their load at

body temperature, and rapidly release the encapsulated drugs within a locally heated tumour

1. INTRODUCTION

23

region. The fast release is due to liposome thermo-responsiveness, which arises from a phase

transition of the constituent lipids and the associated conformational variations in the lipid

bilayers [10-12].

Localised heating can be achieved by various means, including: microwave, radiofrequency

electric current, laser, and HIFU transducer [13]. The sound wave generated by HIFU transducer

is a mature technology which has been adopted in HIFU treatment alone without the use of drugs,

and is found to have very few side effects.

1.2. Objectives of the Study

The delivery of anticancer drugs consists of multiple processes, as the drugs must make their

way into the blood vessels of the tumour, pass through the vessel wall to enter the interstitial

space and be taken up by tumour cells. On the one hand, these processes are dependent on the

physicochemical properties of the drugs, such as diffusivity and drug binding to cellular

macromolecules. On the other hand, the biological properties of a tumour, including tumour

vasculature, extracellular matrix components, interstitial fluid pressure, tumour cell density and

tissue structure and composition, also serve as determinants in these processes. Unfortunately,

the structure and morphology of microvessels in malignant tumours is highly abnormal and

heterogeneous, which create a great barrier for effective delivery of drugs to cancer cells.

The overall aim of this project is to develop a multi-physics model for the simulation of transport

processes of therapeutic agents in solid tumours. The main focus of this project is on the

development of transport models for free drug and drug loaded thermal-sensitive liposomes, with

the following specific objectives:

(1) To develop a mathematical model for the transport of anticancer drugs in their free form;

(2) To determine the optimal administration mode and schedule for maximum accumulation of

drugs in tumour cells;

(3) To develop a mathematical model for the transport of drug-loaded thermosensitive liposomes;

(4) To fulfil objective (3), a bioheat transfer model with HIFU heating needs to be developed;

(5) To examine the effect of different drug delivery modes on drug distribution in solid tumours.

1. INTRODUCTION

24

A research strategy has been developed in order to accomplish these objectives. The

mathematical models are applied to realistic tumour models reconstructed from magnetic

resonance images (MRI). In the basic tumour models, a uniform microvasculature is assumed.

This assumption is subsequently relaxed by incorporating microvasculature heterogeneity

derived from in vivo MRI data, and the effect of microvasculature distribution on drug

concentration in tumour is investigated. Finally, the drug delivery model is coupled with the

bioheat transfer model under HIFU heating in order to study thermal-sensitive liposome-

mediated drug delivery.

1.3. Structure of This Thesis

This thesis includes seven chapters, which are organized in a logical sequence. In Chapter 2, a

comprehensive survey of literature is given for the research to cover tumour properties, drug

delivery modes and drug transport processes. In Chapter 3, mathematical models utilized to

achieve the objectives and their implementations are described. Chapters 4, 5 and 6 present the

findings and analysis of results of drug transport process and treatment efficacy with various

administration modes and biological properties. Chapter 4 compares the bolus injection and

continuous infusion, and Chapter 5 investigates the influence of heterogeneous vasculature and

the size of tumour. Chapter 6 focuses on drug transport under thermal-sensitive liposome-

mediated drug delivery. Finally, conclusions and suggestions for future work are outlined in

Chapter 7.

2. Literature Review

In this chapter, a comprehensive review of the literature on topics related to drug

transport in solid tumour is given. Firstly, the unique characteristics of solid tumour

are described, which serves as an introduction to the microenvironment in tumour.

This is followed by a review of previous studies of drug delivery modes, covering

both medical and mathematical modelling aspects. Finally, the key processes

involved in drug delivery to solid tumour are summarized.

2. LITERATURE REVIEW

26

2.1. Characteristics of Solid Tumour

Different from normal tissues, solid tumours have abnormal characteristics that may strongly

influence the transport of anticancer drugs to and within tumours.

2.1.1. Tumour Vasculature

Microcirculation plays an important role in the delivery of anticancer agents to solid tumours

[14-16]. Small tumours with a diameter less than 2 mm are perfused by blood from their host

normal tissues [17]. With the growth of a tumour, new blood vessels are formed either through

recruitment from pre-existing network of the host tissue or resulting from the angiogenic

response of tumours [18, 19]. Both morphological features and functions of tumour vasculature

differ from those in normal tissues.

Figure 2-1. Normal and tumor vasculature.

A) Normal vasculature: vessels are aligned in a well-organized manner; B) Tumor vasculature: vessels are dilated

with non-uniform diameters and random branching patterns (extracted from [20])

Tumour vasculature becomes tortuous, elongated, and often dilated as a result of tumour growth

(Figure 2-1). The morphological structure of vessels can also change in the growing process of a

tumour, with some features that are rarely present in normal tissues. As shown in Figure 2-2,

trifurcation is a typical pattern which is frequently seen in tumour vasculature. Loops can be

found in both arterial and venous trees in tumours, and there are two types: self-loop and true

loop [21]. Self-loops are planar loops consisting of only two vessels without any other side

branches, whereas true loops are non-planar composed of several vessel segments with many

branches. Polygonal structure exists in the tumour capillary meshwork. Venous convolution is


27

often observed in close proximity to the feeding/draining vessels. The result of this kind of

vascular architecture may lead to plasma skimming and heterogeneous vessel haematocrit levels.

Figure 2-2. Typical architecture of blood vessels in solid tumour (extracted from [21] )

Including (1) trifurcations, (2) self-loop, (3) characteristic polygonal structure of the capillary meshwork, (4) venous

convolutions, and (5) small (20~40 µm) vessels branching off of large (200 µm) vessels

The diameters of arterioles and venules decrease monotonically as branching order increases.

Comparisons presented in Table 2-1 [21-24] show that blood vessels in tumour have a large

diameter, which may exceed the size of anticancer agents (<1~2 µm for free drugs and 100 nm

for liposome particles [25, 26]). Comparisons between unit volume of tumour and normal tissue

show that the length, surface area and fraction of vasculature volume are significantly larger in

tumours.

Table 2-1. Morphological features of tumour microvasculature

Diameter (µm) Length Surface Area Vascular Volume

Fraction (%) Capillary Venules (cm/mm3) (mm

2/mm

3)

In Tumour 5~20 15~70 36 70 50

In Normal Tissue 5~8 12~50 160 20 20

One of the unique characteristics of tumour vasculature is its high leakiness. The abnormal

structure of vessel wall in tumours [27-29] may be one of the reasons. Large inter-endothelial

junctions, increased numbers of fenestrations, vesicles and vesico-vacuolar channels, and a lack

of normal basement membrane are often found in tumour vessels [30, 31]. Perivascular cells

have abnormal morphology and heterogeneous association with tumour vessels. In

correspondence to these structural alterations in the tumour vessel wall, the cut-off size of pores

on tumour microvasculature is in the range of 100~780 nm, varying according to the location and

growth stage of the tumour [26, 32, 33]. Compared to the pore size of 2~6 nm in most normal


28

tissues [34-36] and 40~150 nm in kidney, liver and spleen [37, 38], tumour vasculature with

large pores is favourable for anticancer drugs to pass through, and thereby easing the drug

delivery process.

The distribution of tumour vasculature depends strongly on the location and growth stage of the

tumour. In theory, four regions can be categorized based on the type and function of tumour

vasculature [39]: (1) necrotic region with no functional vasculature; (2) semi-necrotic regions

with capillary, pre- and post-capillary extended; (3) stabilized microcirculation region

characterised mainly by venules, veins and a few arterioles; and (4) tumour advance front with

arterioles, capillaries and venules. Jain simplified this region-categorization as necrotic region,

semi-necrotic region and well-vascularised region along the radial distance [40], as shown in

Figure 2-3.

Necrotic

Region

Semi-

Necrotic

Region

Well

Vascularized

Region

Figure 2-3. Regions in Solid Tumour (extracted from [40] )

In reality, two types of vascular structure can be observed from in vivo images [41]. In tumours

with peripheral vascularisation, the centre is poorly perfused and vessels are located mainly in

the periphery. These may also occur in fully grown tumours that have already invaded into

normal tissues. In tumours with central vascularisation, vessels proliferate from the centre and

form a tree-like structure. In this case, vessel volume in the centre is higher than that in the

periphery of the tumour.

2.1.2. Blood Flow Rate

Given that anticancer agents are carried by the blood stream to the target solid tumour, tumour

blood flow rate may greatly influence the transport of drugs. Under the assumption of steady and


29

fully developed flow and ignoring geometric complexities, the volumetric flow rate in a tumour

vascular network is proportional to the pressure difference between its arterial and venous ends,

and inversely proportional to blood viscosity and geometric resistance [42].

Microvasculature pressure at the arterial ends is found to be similar in both tumour and normal

tissue [43], but pressure in the venular side is significantly lower in a tumour when compared to

normal tissue [17]. Because most blood vessels in tumours are veins whilst arteries mainly exist

in tumour advance front region [41], the driving force for blood flow is greater at the periphery.

This is one possible reason for the heterogeneous distribution of blood flow in tumour [39, 44].

Viscous resistance depends on plasma viscosity and parameters describing the number, size and

rigidity of red blood cells, while geometric resistance is directly associated with the

morphological features of microvessels, such as the number of vessels, branching patterns,

diameter and length [42]. Both of these resistances may be altered by changes in environmental

or internal conditions. Compared to normal tissue, greater resistance is found in tumours owing

to the drainage force from the tumour cells and proteins existing in the extracellular space, as

well as the large vessel diameter and long vessel length.

Table 2-2. Mean blood flow rate in human tumour and normal tissue (ml/g/min) [45].

Brain Lymphatic Breast Uterus Liver

Tumour

0.50 0.45 0.23 0.33 0.12

Normal Tissue 0.57 0.40 0.83 0.13 1.05

A number of studies of blood flow in human tumours and their holding tissues have concluded

that there is a wide range of variations in blood flow rate even for the same type of tumour [45].

Comparisons summarized in Table 2-2 show that the difference in mean blood flow rate between

tumours and normal tissues depend strongly on their location.

2.1.3. Lymphatic Drainage

The lymphatic system is a part of the circulatory system spreading into many organs. Lymphatic

vessels are structurally similar to capillaries, with a wall formed by a single layer of endothelial

cells [46]. However, the lymphatic wall is more permeable than the capillary wall and is not size-

selective [17]. Its main function is to return interstitial fluid back to the blood stream partly due


30

to the low pressure in lymphatics, and to clear extracellular matrix deposits, such as proteins,

from the interstitial space in most normal tissues [47]. The diameter of lymphatic vessels in

tissue is around 40 μm, and the velocity is in the range of 1.4~20 μm/s, with an average value of

8 μm/s [46]. Hence, drugs can be constantly cleared in regions with a high density of functional

lymphatic vessels.

2.1.4. Interstitial Fluid Pressure

The interstitial space of a tumour is complex partly due to the extracellular matrix and non-

uniform vasculature distribution [46]. The pressure in a tumour can exceed atmospheric by

greater than 4200 Pa [46, 48]. This high pressure can be attributed to: (1) insufficient normal

removal of interstitial fluid due to lack of lymphatics; (2) high permeability of tumour

vasculature, and (3) vascular collapse caused by cell proliferation in a confined volume [15].

Models of microenvironment as complex as interstitial space of a tissue require a proper

description of its microstructure. The organization of capillaries in tumours is random owing to

the abnormal morphological features, and the inter-capillary distance is usually 2~3 orders of

magnitude smaller than the length scale for drug transport [49], hence a useful model can be

based on percolation description of porous media governed by Darcy’s law [46].

A theoretical platform has been set by Baxter and Jain [49] for transvascular exchange and

extravascular transport of fluid and macromolecules in tumour based on Darcy’s law describing

the interstitial space as a porous media. Validated by experimental data, their simulations of

tumour microenvironments in both tissue-isolated and subcutaneous tumours indicate that

interstitial fluid pressure in a tumour is higher than in normal tissue with a uniform distribution

in the centre even if in the presence of a necrotic core, and there is a sharp drop of interstitial

pressure at the periphery of the tumour [49-51].

2.2. Modes of Drug Delivery to Solid Tumour

The route and method of drug administrated affect the kinetics of bio-distribution and

elimination, therefore, the effectiveness of chemotherapy [46]. Modern pharmaceutical science

provides many options for drug administration to meet specific requirements in clinical

anticancer treatments.


31

2.2.1. Traditional Administration

Anticancer drug administration traditionally includes oral delivery and intravenous injection.

Because eating is one of the most common acts in daily life, oral intake is always preferred. It is

painless, uncomplicated and self-administrated [46]. However, due to the fact that many drugs

are degraded within the gastrointestinal tract, or cannot be sufficiently absorbed, the treatment

efficacy is seriously compromised.

Drug delivered by intravenous infusion is therefore adopted to offer a more rapid and efficient

therapeutic outcome. With this administration method, nearly 100% of the drug is bioavailable

and the plasma concentration level can also be controlled continuously [46]. The disadvantages,

such as risk of overdose and infection, can also be avoided. Nowadays, intravenous injection is

the most common administration mode in anticancer treatment.

Studies on infusion duration were attempted with an aim to maximise intracellular drug

concentration and reduce side effects. Noticing that high plasma concentration of doxorubicin

may cause serious cardiotoxicity [3, 52], Legha [53] prolonged the infusion duration to 48 hours

or 96 hours in treatments for 21 patients, while the other 30 control patients received standard

intravenous injection with the same clinical dose. It was found that plasma levels of doxorubicin

were reduced by continuous infusion, thereby reducing the risk of cardiotoxicity. Comparison

also showed that the anticancer efficacy was not compromised.

Similar clinical studies were carried out by Hortobagyi [54]. The initial group of breast cancer

patients were treated by doxorubicin with bolus injection while in the other group doxorubicin

was administrated via a central venous catheter over a 48-hour or 96-hour continuous infusion

schedule. Anticancer treatment outcomes revealed that there were no differences between these

two groups, but continuous infusion was less cardiotoxic than bolus injection.

El-Kareh and Secomb [55] suggested that shorter infusion duration might improve treatment

efficacy, and explored it by numerical studies. The anticancer efficacy is reflected by survival

cell fraction based on the predicted peak intracellular concentration over the entire treatment

period. Comparison between bolus injection and continuous infusion of 50, 100, 150 and 200

minutes indicated that infusion duration had a strong influence on the predicted outcome. An

optimal duration for continuous infusion was found to be within 1 to 3 hours.


32

2.2.2. Nanoparticle-Mediated Delivery

In order to reduce the risk of side effects caused by high drug concentration in normal tissues,

cytotoxic drugs are encapsulated in nanoparticles, typically liposomes, before being

intravenously administrated into blood stream [56, 57]. The term “liposome” was firstly

introduced to describe one or more concentric lipid bilayers enclosing an equal number of

aqueous media which can be entrapped in the core during formation [46, 58]. Usually, liposomes

are designed to be in the range of 70 to 200 nm in diameter in order to increase its circulation

time in the blood stream [59]. Decreasing the diameter of liposome to less than 70 nm would

result in 70% of the injected dose being accumulated in liver, while larger liposomal particles

with a diameter greater than 200 nm are more likely to be taken up by the spleen, resulting in a

reduced level in blood [60].

Because of phagocytosis by cells of the reticuloendothelial system, liposomes disappear rapidly

in the blood stream [46]. Hence, modifications on the liposome structure are required.

Polyethylene Glycol (PEG) is now widely adopted as an important component in liposomal layer

in order to prolong the circulation of liposomal nanoparticles in the blood stream. The drug

concentration, therefore, can be maintained at a sufficient high level for effective cell killing [56,

61]. The pharmacokinetics of liposome-mediated doxorubicin in cancer patients was

qualitatively analysed by comparing with the same dose of doxorubicin delivered in free form

[62]. Results have shown clearly that liposome-mediated delivery can reduce plasma clearance

and increase doxorubicin concentration within tumours. Plasma elimination of doxorubicin-

loaded liposomes follows a bi-exponential decay function with half-lives of 2 and 45 hours [62].

In vivo experiments on human prostate carcinoma, which is implanted subcutaneously into nude

Swiss mice, demonstrated that liposome-encapsulated doxorubicin could enhance the therapeutic

efficacy owing to reduced systemic elimination, increase penetration in tumour and prolong

liposome presence with slow drug release [63]. To assess the clinical benefit of liposome-

encapsulated doxorubicin, 297 patients with metastatic breast cancer with no prior chemotherapy

were randomized to receive 60 mg/m2 liposome-encapsulated doxorubicin or conventional

doxorubicin in combination with 600 mg/m2 of cyclophosphamide until disease progression or

unacceptable toxicity was reached [64]. Results showed that liposome-encapsulated doxorubicin

could significantly reduce cardiotoxicity and improve the therapeutic effectiveness.


33

Controlled release is important in making chemotherapy a localised treatment. With a thermo-

sensitive liposomal membrane, in theory, no drug can be released until it is heated beyond its

phase transition temperature [65]. A large number of experimental studies have been carried out

to meet this goal. Thermo-sensitive liposomes sterically stabilized in human plasma have been

developed by Gaber [66], while Unezkazi [67] and Lindner [68] developed improved

formulations to offer both prolonged circulation and thermo-sensitivity. Optimization study on

liposome particle was carried out by Tagami [69] to reduce the release rate at body temperature

while maximising the release rate at mild hyperthermia.

Fast temperature elevation can be obtained by laser, microwaves, radiofrequency electric current

[13] and high intensity focus ultrasound (HIFU) [70]. Ultrasound has been used clinically to

apply thermal therapy non-invasively at targets that are unavailable to other heating methods

[71]. Staruch tested localised drug release with HIFU heating by in vivo experiments on rabbits

[72, 73]. In their experiments, a HIFU beam was scanned in a circular trajectory to heat a given

region in normal thigh [72] and tumour [73] to 43oC lasting for 20~30 minutes. Localised

heating is controlled by MRI thermometry. Encapsulated doxorubicin with thermo-sensitive

liposome was intravenously administrated during hyperthermia. Comparison between heated and

unheated regions indicated that MRI-controlled HIFU hyperthermia could enhance local drug

delivery.

Temperature elevation in a liver tumour was predicated by numerical studies in Sheu’s work [74].

Based on a patient-specific liver model, the large hepatic artery was modelled explicitly to

investigate the impact of blood flow on temperature elevation. However, neither the real

geometry of hepatic tumour nor drug release was included in this study.

Researchers also investigated HIFU heating modes in order to achieve a homogeneous

temperature profile in tumour. Based on a 2-D idealized circular model, variables in fast

scanning method with a short sonication duration for HIFU treatment were examined [75].

Results showed that fast scanning method could produce a planned lesion regardless of its

scanning path. The final temperature profiles were highly dependent on the applied power and

tissue perfusion. Theoretical and experimental studies [76, 77] suggested that signal point fixed-

gain proportional-integral control coupled with multiple heating points along a rapid scanned

trajectory could result in a homogeneous temperature profile in a given region.


34

2.2.3. Polymer Implant

Apart from nanoparticle-mediated delivery, a polymer implant is another delivery method

developed to overcome the lack of targeting in common methods. The anticancer drug is firstly

dissolved in polymers, and then the drug needs to diffuse through the polymer after being

implanted and finally dissolve into the extracellular space of tumour. A number of polymeric

materials have now been approved for clinical use. The most extensive type is non-degradable

hydrophobic polymers. The development of biodegradable polymers with the advantage that no

residual material remains in the tissue has enabled more anticancer agents with low diffusivity

to be delivered to solid tumours [46]. However the design of these polymers is complex since

large quantities of potentially harmful products may be released in the body.

Wang [78] showed that implanting IgG-loaded polymer in the cavity left after surgical removal

of a tumour may improve the anticancer efficacy. However, Teo [79] found that surgical removal

of the tumour alone could only enhance drug delivery in bis-chloroethylnitrosourea (BCNU)

treatment in limited time. Intratumoural administration with optimization is capable to improve

treatment efficacy and reduce drug concentration in normal tissue [80, 81].

2.3. Drug Transport in Solid Tumour

Systemic administration is the main delivery method for chemotherapeutic agents. Following its

administration into the blood stream, anticancer drug experiences sequential processes before

reaching the targeted tumour cells. These processes (shown in Figure 2-4) are described in detail

below.

Protein

Cells

Blood

Vessel

Interstitial

Fluid

Drug

Figure 2-4. Schematic diagram of drug transport


35

2.3.1. Transport within Blood Vessel

After administration, therapeutic agents begin to circulate in the systemic circulation to all over

the body. Drug transport within blood stream is determined by drug properties and the in vivo

environment. Drugs that are transported via the blood may disperse into tissue when passing by.

In addition, kidney and other organs can eliminate drugs from the circulation system by filtering

plasma continuously, and thereby compromising the treatment effect. This function of renal

excretion is known as plasma clearance, which may vary according to the drug type [82-84].

Rapid Peripheral

Compartment (V2)

Central

Compartment

(V1)

Slow Peripheral

Compartment (V3)

I(t)

Figure 2-5. Schematic diagram of three-compartment model

Pharmacokinetics is classically represented by compartment models describing drug plasma

concentration as a function of time, I(t). A human body can be represented as three

compartments with the volume distribution at steady state shown in Figure 2-5: the central

compartment (V1) for blood plasma, rapid (V2) and slow (V3) peripheral compartments

corresponding to dense and sparse vessel volumes, respectively. Drugs in the central

compartment can be cleared out, and there are two-way exchanges between the central and

peripheral compartments, as shown in Figure 2-5. A two-compartment model has also been

adopted in mathematical modelling of the delivery of anticancer agents, which consists of a

central compartment and periphery compartment, with inter-compartmental exchange taking

place in both directions.

Plasma concentration is associated with plasma clearance, dosage and administration mode. The

pharmacokinetics of doxorubicin has been evaluated in the plasma of 12 breast cancer patients

receiving no prior treatment [85]. Drug concentration was obtained by testing plasma samples

collected at various times after administration. The successive half-lives were 0.08 hour, 0.822

hour and 18.9 hour, indicating that a three-component exponential function would be better

suited for doxorubicin concentration in blood plasma.


36

In addition to red blood cells, white blood cells and platelets, blood also contains a range of

proteins. Extensive bindings between these proteins and drugs have been observed in clinical and

experimental conditions. Plasma pharmacokinetics of doxorubicin with a 15 minutes continuous

infusion were studied in 10 breast cancer patients [86]. By applying high performance liquid

chromatography, Green found that 75±2.7% of the total plasma concentration of doxorubicin

was bound with protein over the whole range of observed concentrations. This high rate of

binding can effectively reduce the plasma concentration of free doxorubicin and affect the

amount of drug reaching tumour cells. On the other hand, the retention of drug in blood can be

extended because of the low plasma clearance rate of protein [17].

2.3.2. Transport through Blood Vessels

Being transported within the circulatory system, some drugs can be extravasated to the interstitial

space owing to the high permeability of tumour vasculature. Drug transport through the

vasculature wall depends on convection and diffusion.

Drug transport by convection is associated with fluid flux across the vessel wall. This fluid

movement is determined by hydrostatic and osmotic pressure difference between the lumen and

interstitium. Drug transport by diffusion is driven by the concentration gradient between blood

and interstitial fluid, and is also related to the permeability and area of blood vessels for drug

exchange. This implies that drug can also be transported back to blood when the gradient

reverses. Although drugs can be carried by endothelial cells, transcytosis has been found to play

a minor role when compared with convection and diffusion [17].

The pore model is widely adopted in numerical study of drug transport through blood vessels.

Deen [87] reviewed studies of the hindered transport through pores on membrane that were

aimed at predicting applicable transport coefficients from fundamental information, such as the

size, shape and electrical charge of the solutes and pores in liquid-filled pores. The pioneering

work on drug transport in solid tumour was conducted by Baxter and Jain. Employing the pore

model described in Deen’s work [87], they developed a general mathematical platform for trans-

vascular and interstitial transport of fluid and macromolecules [49, 50, 88, 89].

Through application to an idealized tumour core model, Eikenberry [90] investigated the

delivery of soluble doxorubicin from a capillary to its surrounding tumour mass to examine the


37

penetration of doxorubicin under different dosage regimens and microenvironment in solid

tumour. Drug infusion duration was found to be an important factor in determining the spatial

profile of tumour cell killing, and extending the infusion duration or fractionating the dose would

help improve the treatment effect. Numerical results showed that only a limited amount of

doxorubicin could be delivered into solid tumours.

The application of a pore model was extended by El-Kareh and Secomb [55] to study liposome-

mediated delivery of doxorubicin to solid tumours. A mathematical model was developed and

applied to compare the performance of various administration modes of both free form and

liposomal encapsulated doxorubicin. The treatment outcome was predicated based on peak

intracellular concentration without considering the dynamics of tumour cell density. Their

simulation results indicated that continuous infusion with an optimal infusion duration yielded

the best treatment efficacy, and there was a clear advantage of thermo-sensitive liposomal

delivery at some doses but only if the blood was not significantly heated.

2.3.3. Transport in Interstitial Space

Following extravasation, drugs move in the interstitial space to reach tumour cells. The main

pathways for drug movements are also attributed to convection and diffusion, which are highly

dependent on interstitial fluid pressure and interstitial fluid velocity within tumour. Drug

convection in interstitium is determined by hydraulic conductivity and pressure gradient, and

diffusion depends on the concentration gradient and diffusion coefficient. Diffusion within a

tissue is slower than diffusion through a medium filled with interstitial fluid, due to the presence

of cells [46]. The composition of the extracellular space also affects the diffusivity of compounds

through tissue, especially for macromolecules which are mostly influenced by the presence of

proteins in interstitial fluid. The fact that diffusivity also depends on the architecture of tissue [46]

means that the diffusion coefficient for the same drug can be different in tumour and normal

tissue.

In the work of Netti [91], the tumour was modelled as a poroelastic material. The aim was to

determine the mechanisms regulating interstitial fluid pressure and develop methods to overcome

the interstitial fluid pressure barrier in drug delivery. Experimental and numerical results

demonstrated that higher uptake of macromolecular would result from periodic modulation of


38

blood pressure rather than acute or chronic increase of intravascular pressure. Hence pressure

elevation caused by drug administration would have a limited effect on drug delivery.

Goh [51] develop a 2-D simulation platform to investigate the spatial and temporal variation of

doxorubicin concentration in hepatoma. The model geometry was reconstructed from patient-

specific CT images, including the necrotic core, viable tumour and its surrounding normal tissue.

Increase in vascular pressure caused by intravenous bolus injection was considered, and the

predicated results showed that interstitial fluid pressure stabilized fairly quickly. The results also

indicated that diffusion was the dominant transport mechanism. It was concluded that

doxorubicin transport to solid tumour was strongly dependent on the vascular exchange area,

doxorubicin cellular metabolism and DNA binding kinetics.

The effect of transient interstitial fluid flow on drug delivery in brain tumour was addressed by

Teo [79]. After surgical removal of the necrotic core, polymer wafers that contained bis-

chloroethylnitrosourea (BCNU) were implanted in the cavity. The mathematical platform

established by Goh [51] was applied to simulate the intra-tumoural release of BCNU and its

penetration through resection cavity to the tumour. The interstitial fluid flow was found to reach

equilibrium in less than 3 hours, and surgery could only enhance drug delivery temporally.

Owing to the presence of a variety of proteins and nucleic acids in the interstitial space,

extensive binding may take place. Experimental studies have shown that binding can delay drug

transport [92-94]. In an experiment on thyroid cancer cells, 5-fluorouracil, cisplatin and inulin

were found to be evenly distributed in cell spheroids within 15 minutes [17, 93, 94], while

doxorubicin, methotrexate and paclitaxel, which bind to macromolecules, remained in the

periphery of spheroids [95-97]. Since a tumour is always surrounded by normal tissues and

interstitial fluid velocity is higher at the periphery of tumour than in the centre, drug exchange

can also take place between tumour and normal tissues.

Based on a computational reaction-diffusion model, Tzafriri [80] examined drug release from

intra-tumourally injected microspheres and drug transport in an idealized tumour model

accounting for binding in tumour interstitial fluid. Their simulation results suggested that

extracellular drug concentration was spatially heterogeneous in the initial period following drug

administration. It was concluded that increasing the duration of drug release might prolong the


39

exposure duration of anticancer drug but lower the plateau drug concentration. This implied that

intratumoural drug release could be optimized to improve treatment efficacy.

Baxter and Jain [88] developed a general theoretical framework to examine the effect of binding

on concentration profiles of macromolecules. A high binding affinity was predicted to be most

efficacious except for certain conditions of necrotic cores and low vascular permeability.

Doxorubicin was also found to bind extensively with proteins, mainly albumin, in both plasma

and interstitial fluid [55]. Due to the lack of in vivo experimental data, Eikenberry [90] assumed

that doxorubicin binding and unbinding in both plasma and interstitial fluid shared the same rates,

which were obtained based on experimental data [86]. Coupled with other drug transport

processes including transvascular and interstitial movement, drug binding was investigated in a

tumour core model. Their simulation results showed that most of the doxorubicin in extracellular

space existed in bound form.

2.3.4. Transport to Lymphatic System

The lack of a functional lymphatic system has been observed in solid tumours [49, 50]. The

limited existence of the lymphatic system can significantly reduce the clearance of large

molecular weight drugs and hence increase their retention in interstitial fluid. These are typically

liposome, nanoparticles and macromolecular drugs. On the other hand, without effective

drainage of interstitial fluid from the extracellular space, interstitial fluid pressure will increase,

which will in turn affect the drug exchange between blood and interstitial space.

Based on previous work on interstitial fluid pressure and drug convection, Baxter and Jain [50]

used a similar model to examine the effect of lymphatics on interstitial fluid pressure distribution

and concentration distribution of macromolecules (e.g. antibody). Their results suggested that

substances could be rapidly removed by lymphatics, if present in tumour, which would result in

much lower concentration levels.

2.3.5. Transport through Cell Membrane

After migrating in the interstitial space, drug can pass through cell membrane to enter tumour

cells. The main mechanisms for molecular transit across membranes are passive diffusion,

diffusion through channels, facilitated and active transport which involves the participation of


40

transmembrane proteins that bind a specific solute [46]. Drug transport through cell membrane is

a dynamics process, with influx and efflux occurring simultaneously [46, 90]. Dordal [98]

monitored cellular uptake of fluorescence-labelled doxorubicin in human lymphoma cells by a

flow cytometry assay with a rapid-injection system. The experimental data was then analysed by

a pharmacokinetic model to find the relation between drug concentration and cellular uptake.

Tumour cells can become resistant to anticancer drugs. One of the complex resistance

mechanisms is related to a 1.7×105

Da membrane protein named P-glycoprotein (P-gp). Being

insensitive to a wide variety of chemicals, P-gp is capable of actively transporting certain types

of anticancer drugs, including doxorubicin and vincristine. Luu [99] developed a mathematical

model to capture the efflux of doxorubicin resulting from drug mediated P-gp induction, and

they concluded that their mathematical model could be utilized to optimize dosing strategies in

order to minimize P-gp induction and thereby improving treatment efficacy.

El-Kareh and Secomb [55] included the dynamic process of cellular uptake and efflux in a

symmetric tumour model to investigate the influence of administration modes. Friedmen [100]

adopted a simplified model to fit cellular uptake rate data that showed saturation in both intra-

and extra-cellular concentrations. Kerr’s [101] experiments provided the experimental support to

this mathematical model. Cell uptake and efflux were analysed by human non-small cell lung

tumour cells in monolayer culture, and the relation between intracellular drug concentration and

cytotoxicity was obtained. A similar model was improved by Eikenberry [90] and then adopted

by Liu [102] to investigate the effects of ATP-binding cassette transporter-based acquired drug

resistance mechanisms at the cellular and tissue scale.

2.3.6. Cell Killing

After phagocytosis by tumour cells, drugs may destroy the genes that cause cells to divide or

interrupt the chemical process involved in cell division, and thereby killing cells. The rate of cell

killing has been found to be strongly dependent on extracellular and intracellular drug

concentrations, and several models have been proposed to describe this relationship.

The cytotoxicity of drug was firstly assumed to be a function of area under the extracellular

concentration-time curve (AUCe) [103]. Ozawa [104] derived a cell kill kinetics equation for cell

cycle phase-non-specific agents, and tested the equation by fitting to experimental results based


41

on Chinese hamster V79 cells. It was concluded that the cell killing action of cell cycle phase-

non-specific drugs could be described by a pharmacodynamics model with the concentration-

time product. Eichholtz-Wirth [105] treated Chinese hamster cells and Hela cells with

doxorubicin, and found the surviving cell fraction was proportional to the product of

extracellular drug concentration and exposure time. The experimental data also showed a non-

linear relation between extra- and intra-cellular concentrations.

Although there is evidence that drug can kill cells without entering them, cell death is believed to

be mainly dependent on intracellular concentration [1, 106]. Alternative models including

intracellular drug concentration and cellular pharmacodynamics have been proposed. A cellular

pharmacodynamics model for cisplatin was proposed by El-Kareh and Secomb [107] to relate

cell killing to the peak value of intracellular concentration over the entire treatment period. The

kinetics equations were modified for doxorubicin to reflect survival cell fraction [55, 90].

However, this model is incapable to tracking the dynamic variation of tumour cell density.

Given that chemotherapy is a dynamic process including cell proliferation, physical degradation

and cell killing caused by drugs, a dynamic model was suggested to evaluate the temporal

variation of cytotoxic effect of anticancer drugs [102]. Lankelma [108] proposed a

pharmacodynamics model for doxorubicin, where cell killing was described as a function of

intracellular concentration history. Drug accumulation history in cells was calculated using

cellular drug transport parameters derived from doxorubicin uptake and efflux measurements on

MCF-7 cells attached to culture dishes. Eliaz [109] developed a model taking into account the

cell proliferation rate, the cell killing rate, the average intracellular concentration and a lag time

for cell killing. The model was applied to free doxorubicin and doxorubicin encapsulated in

liposome targeted to melanoma cells in culture.

2.4. Summary

A detailed review of tumour characteristics, drug delivery methods and transport processes

indicates that all these three aspects depend on and interact with each other during anti-tumour

treatment. Although previous studies summarized in this chapter have addressed the complex

interplays among these factors to a certain extent, a systematic study that accounts for all these

physical and biological aspects is lacking. This is what the current project aims to achieve, and


42

the integrated mathematical model developed in this project will serve as a platform to compare

and optimise different strategies in chemotherapy.

3. Mathematical Models and Numerical

Procedures

In this chapter, a detailed description of mathematical models and their

implementations are given. The mathematical equations are described first; these

include equations governing the interstitial fluid flow in tumour and normal tissues,

mass transport and cellular uptake of anticancer agents, and heat transfer in response

to the application of HIFU for activation of drug release from thermo-sensitive

liposomes. Model parameters as well as relevant physical and transport properties are

then discussed. Finally, numerical methods to implement the mathematical models

are presented.

3. MATHEMATICAL MODELS AND NUMERICAL PROCEDURES

44

3.1. Mathematical Models

Drug properties are the key factors in anticancer treatments. In addition, drug transport also

depends on the interstitial fluid flow in tumour and its holding tissue, and the properties of

tumour cells have impacts on the cell killing as well. To include these relationships, the current

modelling platform consist of descriptions of interstitial fluid flow, transport for none-

encapsulated and thermo-sensitive liposome-mediated drugs, pharmacodynamics of tumour cells

and bioheat transfer model introduced by high intensity focus ultrasound heating for thermo-

sensitive liposome-mediated drugs only. Given the mentioned fact that the large size difference

exist between a tumour and its vasculature, a commonly adopted approach for study on drug

transport is treating the vasculature as a source term in the governing equations, without

considering its morphological features. The main assumptions are as follows: (1) the interstitial

fluid is incompressible, Newtonian fluid with homogeneous properties in tumour and

surrounding normal tissue, respectively; (2) all tumour cells are stationary, identical and alive

initially; (3) the simulation duration is significantly shorter than the timescale for tumour growth,

and thereby the physiological parameters and geometry are independent of time.

The mathematical models consist of the mass and momentum conservation equations for

interstitial fluid flow, mass transfer equations for the free and bound drug, transport equations for

liposome encapsulated drug, as well as equations describing the intracellular drug concentration

and pharmacodynamics. First, the interstitial fluid flow equations are solved to provide the basic

biomechanical environment for drug transport. This is followed by solution of the mass transfer

equations for drug transport, which is described schematically in Figure 3-1 (a) and (b) for direct

infusion, and thermo-sensitive liposome-mediated delivery of doxorubicin, respectively. Briefly,

the tumour region consists of three compartments: blood, extracellular space and tumour cells.

Within each compartment, letters F, B and L represent free, bound and liposome encapsulated

doxorubicin, respectively.

The dynamics process in drug delivery includes association/disassociation with protein at the rate

kas and kds respectively, drug exchange between blood and extracellular space, and influx/efflux

of drugs from extracellular space to tumour cells. The rate of cell killing is governed by a

pharmacodynamics model based on the predicted intracellular concentration of anticancer drugs.

In the case of thermo-sensitive liposome-mediated drug delivery, an additional equation


45

describing the transport of liposomes is needed. The detailed descriptions of each section of

mathematical models are listed below.

F

B

F

B

F

Blood Extracellular Space Tumour Cell

Cell

Killing

kas kds kas kds

influx

exfflux

clearance

exch

ange

(a)

F

B

L

F

B

L

Blood Extracellular Space

F

Tumour Cell

Cell

Killing

krel

kas kds kas kds

influx

exfflux

clearance

clearance

ex-

change

ex-

change

krel

(b)

Figure 3-1. Drug transport with (a) continuous infusion of drug in its free form, and (b) thermo-sensitive liposome-

mediated drug delivery

3.1.1. Interstitial Fluid Flow

The mass conservation equation for interstitial fluid is given by

lyv FF v (3-1)

where v is the velocity of the interstitial fluid, Fv and Fly are the source and sink terms,

accounting for the gain of interstitial fluid from blood vessels, and fluid absorption rate by the

lymphatics, respectively. Fv and Fly can be determined by Starling’s law


46

ivTivvv ppV

SKF (3-2)

where Kv is the hydraulic conductivity of the microvascular wall, S/V is the surface area of blood

vessels per unit volume of tumour tissue, pv and pi are the vascular and interstitial fluid pressures,

respectively, σT represents the average osmotic reflection coefficient for plasma protein, πv is the

osmotic pressure of the plasma, and πi is that of interstitial fluid.

The lymphatic drainage, Fly, is related to the pressure difference between the interstitial fluid and

lymphatics.

lyi

ly

lyly ppV

SKF (3-3)

where Kly is the hydraulic conductivity of the lymphatic wall, Sly/V is the surface area of

lymphatic vessels per unit volume of tumour tissue, and ply is the intra-lymphatic pressure.

The tumour and its surrounding tissues are treated as porous media, which can be justified as the

inter-capillary distance (33-98 µm [21, 110]) is typically 2-3 orders of magnitude smaller than

the length scale for drug transport. The corresponding momentum equation is given by

Fτvv

v

ip

t

(3-4)

where the stress tensor τ is expressed as

Ivvvτ )(3

2])([ T

(3-5)

where ρ and μ is the density and dynamic viscosity of interstitial fluid, respectively. I is the unit

tensor. The last term in equation (3-4), F, represents the Darcian resistance to fluid flow through

porous media and is given by

vvCvF 2

1W (3-6)

and W is a diagonal matrix with all diagonal elements calculated as


47

1 W (3-7)

where C is the prescribed matrix of the inertial loss term, and κ is the permeability of the

interstitial space. Since the interstitial fluid velocity is very slow (|v|<<1) [49], the inertial loss

term can be neglected when compared to the Darcian resistance. In addition, the interstitial fluid

is treated as incompressible with a constant viscosity. Hence, equation (3-6) can be reduced to

vF W (3-8)

3.1.2. Drug Transport

Drug transport is governed by the convection-diffusion equations for the free and bound drug in

the interstitial fluid. Free doxorubicin concentration in the interstitial fluid (Cfe) is described by

ifefefe

feSCDvC

t

C

2 (3-9)

where Dfe is the diffusion coefficient of free doxorubicin. The source term, Si, is the net rate of

doxorubicin gained from the surrounding environment, which is given by

ubvi SSSS (3-10)

here Sv, Sb and Su represent the net rate of doxorubicin gained from the blood/lymphatic vessels,

association/dissociation with bound doxorubicin-protein and influx/efflux from tumour cells,

respectively.

flfpv FFS (3-11)

where Ffp is the doxorubicin gained from the blood capillaries in tumour and normal tissues, and

Ffl is the loss of doxorubicin to the lymphatic vessels per unit volume of tissue. Using the pore

model [49, 50, 87, 89] for trans-capillary exchange, Ffp and Ffl can be expressed as

1

Pe1

fPe

f

eCC

V

SPCσFF fefpfefpdvfp (3-12)

felyfl CFF (3-13)


48

where Cfp is the concentration of doxorubicin in blood plasma, σd is the osmotic reflection

coefficient for the drug molecules, and Pfe is the permeability of vasculature wall to free

doxorubicin. Pef is the trans-capillary Peclet number defined as

V

SP

F

fe

dv

1Pef (3-14)

The net doxorubicin gained due to protein binding and cellular uptake is governed by

feabedb CkCkS (3-15)

ζDDS ccu (3-16)

where Dc is the tumour cell density, ka and kd are the doxorubicin-protein binding and

dissociation rates, respectively.

The convection-diffusion equation for bound doxorubicin (Cbe) is similar to equation (3-9)

except for the source terms.

bbebe

2

bebebe SFCDCt

C

v (3-17)

where Dbe is the diffusion coefficient of the bound doxorubicin-protein, and Fbe accounts for the

gain of bound doxorubicin from blood vessels, which is given by

1

Pe1

bPe

b

eCC

V

SPCFF bebpbebpdvbe (3-18)

where Pbe is the permeability of the vessel wall to bound doxorubicin, and Cbp is the bound

doxorubicin concentration in the plasma. The trans-capillary Peclet number is

V

SP

F

be

dv

1Peb (3-19)

Since only unbound doxorubicin can pass through the cell membrane [55, 90], the rate of cellular

uptake is a function of free doxorubicin concentration in the interstitial fluid.


49

t

Ci (3-20)

efe

fe

kC

CV

max (3-21)

ii

i

kC

CV

max (3-22)

where Ci is the intracellular doxorubicin concentration, and Vmax is the rate of trans-membrane

transport, ζ and ɛ are cellular uptake and efflux functions, ke and ki are constants obtained from

experimental data, and φ is the volume fraction of extracellular space.

3.1.3. Thermo-Sensitive Liposome-Mediated Drug Transport

Additional equations are required to describe the transport of liposome encapsulated doxorubicin,

in order to solve separately the encapsulated drug concentration and released drug concentration.

Equations for the transport of released drug, including those for free drug concentration in

plasma and interstitial fluid, bound drug concentration in plasma and interstitial fluid as well as

intracellular concentration, are described using the same equations given in the preceding section.

Similar to equation (3-9), the liposome encapsulated drug concentration in the interstitial fluid

(Cle) is governed by

llellele SCDCt

C

2v (3-23)

where Dl is the diffusion coefficient of liposomes. The source term, Sl is the net rate of liposomes

gained from the surrounding environment, which is given by

rlpl SSS (3-24)

Here, Slp represents the liposomes gained from plasma, and Sr represents released drug in the

interstitial fluid. Slp can be calculated by

lllplp FFS (3-25)


50

where Flp is the liposomes gained from the capillaries in tumour and normal tissues, and Fll is the

loss of liposomes through the lymphatic vessels per unit volume of tissue. Values for Flp and Fll

can be obtained using the pore model described by Equations (3-12) to (3-14) but with transport

properties corresponding to the thermo-sensitive liposome.

The amount of released doxorubicin in the interstitial fluid, Sr, is given by

lerelr CkS (3-26)

where krel is the release rate of doxorubicin from liposome.

The free doxorubicin concentration in blood plasma (Cfp) is governed by

bpdfpafpfpfp

B

Tr

fpCkCkCCLF

V

VS

t

C

(3-27)

where Ffp represents the free doxorubicin crossing the capillary wall into the interstitial fluid. VT

and VB are tumour volume and blood plasma volume, respectively. CLfp is the plasma clearance

of drug. ka and kd are the association and disassociation rates with proteins.

The bound doxorubicin concentration in blood plasma (Cbp) is given by

bpbpbe

B

Tbpdfpa

bpCCLF

V

VCkCk

t

C

(3-28)

where CLbp is the plasma clearance of bound doxorubicin.

The free doxorubicin concentration in interstitial fluid (Cfe) is described by Equation (3-9),

except that the source term contains an additional contribution from Sr, as described below.

rubvi SSSSS (3-29)

Expressions for the terms on the right hand side have been given previously (see equations (3-11)

to (3-16) and (3-26)).


51

3.1.4. Pharmacodynamics Model

The change of tumour cell density with time is described by a pharmacodynamics model as

given below [109].

2

50

maxcgcpc

i

ic DkDkDCEC

Cf

dt

dD

(3-30)

The first term on the right hand side represents the anticancer effect, where fmax is the cell-kill

rate constant and EC50 is the drug concentration producing 50% of fmax. kp and kg are cell

proliferation rate constant and physiological degradation rate, respectively. In this study, cell

proliferation and physiologic degradation are assumed to have reached equilibrium at the start of

each treatment.

3.1.5. Heat Transfer under High Intensity Focused Ultrasound (HIFU) Heating

In localized heating in anti-tumour therapy, both blood and tissue including the tumour and its

holding tissue are heated. Energy balances for tissue and blood are illustrated in Figure 3-2 (a)

and (b), respectively.

TISSUE

Qulltrasound

Qexchange

with blood

(a)

BLOOD

Qultrasound

Qexchange

with tumour

Qblood flow (Tb)Qblood flow (Tn)

(b)

Figure 3-2. Schematic representation of heat transfer under HIFU heating in (a) tumour and (b) blood

The temperature (T) of tissue and blood can be calculated by solving the following energy

balance equations [74, 111]:

tbtbbbttt

tt qTTwcTkt

Tc

2 (3-31)

bnbbbbbtbbbbbb

bb qTTwcTTwcTkt

Tc

2 (3-32)


52

where ρ is the density, c is the specific heat, k is the thermal conductivity, and w is the perfusion

rate of blood flow. q represents the ultrasound power deposition, and subscripts b and t represent

blood and tissue, respectively. Tn is the normal body temperature, which is assumed to be 37 ˚C.

In reality, heating of biological tissue by HIFU also contains the effect of acoustic non-linearity

[112, 113]. However, studies show that the non-linear wave propagation can be ignored if a focal

intensity is within the range of 100~1000 W/cm2 and the peak negative pressure in the range of

1~4 MPa [74, 114]. The ultrasound power deposition per unit volume is assumed to be

proportional to the local acoustic intensity Ia as follows

aIq 2 (3-33)

where α is the absorption coefficient, and the intensity Ia is defined as:

a

aa v

pI

2

2

(3-34)

va is the speed of ultrasound, and pa is the acoustic pressure which is represented by the real parts

of the following equation

aaca vikp (3-35)

here kac represents the acoustic velocity potential, and is given by

2a

ac vk (3-36)

ω is the angular frequency, and λ is the wave length of ultrasound.

For a spherical concave radiator S (Figure 3-3) with a uniform normal velocity, if the diameter or

width of a slightly curved radiator is large compared to the wave-length, the resulting acoustic

velocity potential, Ψ, in a restricted region can be written as [115]

2

0 011

1

2,

bdik

ddRResR acu

(3-37)

where u is the normal velocity of a slightly curved radiating surface, which is represented by


53

tieu

0u (3-38)

here u0 is a constant and t is time. In Equation (3-37), d is the distance from a point on the

radiator to a field point, which is given by

2

11

2 cossin2 zRRrRd (3-39)

2cos21 bhRz (3-40)

22

111 41 ARRr (3-41)

Referring to Figure 3-3, the ultrasound radiator is represented by the thick lines and the heated

point is Q. A and a are the radius of curvature and radius of the circular boundary, respectively, h

is the depth of the concave surface and b is the chord from the transducer centre O to the radiator

boundary.

Q1Q

r

C

Aa

b

h

R1

θ1 θ αα

x

O

R

d

S

Figure 3-3. Dimensions and coordinates of a spherical radiator [115].

3.2. Model Parameters

Since the growth of tumour and normal tissues is ignored, all the geometric and transport

parameters used in this study are assumed to be independent of time. Values adopted for these

are summarized in Tables 3-1, 3-2 and 3-3 at the end of this Chapter for parameters related to the

tissue, liposome and doxorubicin, respectively. Justifications for the choices of some of the

parameters are given below.


54

3.2.1. Tissue Related Transport Parameters

Blood vessel surface area to tissue volume ratio (S/V)

The ratio of surface area of blood vessel to tissue volume has a direct influence on the amount of

anticancer drug in the interstitial fluid. Its value depends strongly on the type of tissue and stage

of tumour growth [116]. Pappenheimer et al. measured this in normal tissues [117], while Baxter

and Jain [49] recommended using 70 cm-1

and 200 cm-1

for normal and tumour tissues,

respectively. The effect of this parameter on anticancer efficacy and its derivation from in vivo

MR images will be discussed in Chapter 5.

Tumour cell density (Dc)

The volume fraction of extracellular space in a tumour tissue ranges from 0.2 to 0.6 [118].

Assuming an average tumour cell diameter, Eikenberry [90] estimated that tumour cell density

ranged from 9.55×1015

to 1.53 ×1016

cells/m3. During chemotherapy treatment, tumour cell

density may change as a result of drug uptake, cell proliferation and physiological degradation.

3.2.2. Drug Related Transport Parameters

Vascular Permeability (P)

Vascular permeability coefficient measures the capacity of a blood vessel (often capillary in

tumour) wall to allow for the flow of substances in and out of the vasculature. The structure of

vessel wall and the molecular size of the transported substance are key determinants of

permeability [110].

(1) Free & bound doxorubicin

Estimates of this parameter reported in the literature usually correspond to ‘effective

permeability’, which is on the order of 10-7

m/s for albumin in both tumour and normal tissues.

For Sulforhodamine B (MW=559 Da), its permeability in normal tissue is 3.4×10-7

m/s [119].

Patent is another agent with a similar molecular weight (566 Da) to that of doxorubicin, and its

permeability is 3.95×10-7

m/s in normal tissue of male frog [120]. Eikenberry [90] assumed the

effective permeability of free doxorubicin in tumour to be in the range of 8.1×10-7

and 3.7×10-6

m/s, whereas Ribba et al. [121] used 3.0×10-6

m/s in their simulation work. Compared with


55

normal tissues, Gerlowski and Jain [122] found the vessel wall permeability to be 8 times higher

in tumour tissues. Using MW for interpolation, Goh [51] assumed the permeability of

doxorubicin in tumour tissue to be nearly 8 times higher than in normal tissue. In the present

study, values of permeability in tumour and normal tissues are assumed to be 3.0×10-6

m/s and

3.75×10-7

m/s, respectively. Wu et al. [119] measured the permeability of albumin

(corresponding to albumin-bound doxorubicin) in both tumour and granulating tissues, and found

the corresponding vasculature permeability to be 7.8±1.2×10-9

m/s and 2.5±0.8×10-9

m/s,

respectively. In this thesis, their average value is adopted as the permeability of albumin-bound

doxorubicin.

(2) Liposome encapsulated doxorubicin

The permeability of polyethylene glycol coated liposomes of 100 nm through tumour capillaries

was measured at 37 o

C by Yuan et al. [26] and Wu et al. [25] as 2.0×10-10

and 3.42±0.78×10-9

m/s, respectively. In normal granulation tissues permeability of the same liposomes was

8.0~9.0×10-10

m/s at the same temperature [25, 26].

Diffusion Coefficient (D)

Diffusion coefficient is the constant of proportionality between the molar flux owing to

molecular diffusion and the gradient in the concentration of the species, which is also known as

the driving force for diffusion.

(1) Free and bound doxorubicin

Diffusion coefficient is related to the molecular weight (MW) of the substance [15]. The

relationship between diffusion coefficient in water and molecular weight can be represented by

[15, 123]

d2a

d1w MWaD

(3-45)

This equation is used to fit the diffusion coefficients of common anticancer agents in water [124]

(Figure 3-4). Since doxorubicin has a MW of 544 Da, its diffusion coefficient in water at 37 oC

could be estimated as 3.83×10-10

m2/s. It has also been found that diffusion coefficient in

neoplastic tissue deviates from free diffusion in water [118]. For MWs in the range of 376 Da and


56

66,900 Da, the ratio of diffusion coefficient in tumour (D) to that in water (DW) was found to be

linearly related to MW

d4d3w

aMWaD

D (3-46)

Using equation (3-46), the ratio for doxorubicin is 0.8885 and the diffusion coefficient of

doxorubicin in tumour is 3.40×10-10

m2/s (Figure 3-4 (b)).

At 37 oC, the relationship between diffusion coefficient (D) and MW in normal tissues may be

obtained from in vitro experimental data [125].

690003210778.175.08

MWMWD (3-47)

Based on equation (3-47), the diffusion coefficient of doxorubicin in normal tissues is calculated

as 1.58×10-10

m2/s.

50 100 150 200 250 300 350 400 450 500 550 600 650

2

3

4

5

6

7

8

9

10

11

12

Dif

fusi

on C

oef

fici

eng i

n W

ater

(10

-10 m

2/s

)

Molecular Weight (Da)

Data from Literature

Fitting Curve

(a)

0 10000 20000 30000 40000 50000 60000 70000

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

DT/D

W

Molecular Weight (Da)

Data from Literature

Fitting Curve

(b)

Figure 3-4. Estimation of doxorubicin diffusion coefficient in tumour tissue. (a) The diffusion coefficient of

anticancer drugs in water as a function of molecular weight, using equation (3-45) [15, 123]. For the best fitted

curve for experimental data in reference [124], ad1 = 380.44, ad2 = 0.73. (b) The relation between diffusion

coefficient ratio (tumour to water) and molecular weight. For the best fitted line using equation (3-46) to fit

experimental data from reference [118], ad3 = -8.55948 10-6

, ad4 = 0.89317.

Doxorubicin can bind with high molecular weight proteins (e.g. albumin [90]) in the interstitial

fluid. In such a case, the diffusion coefficient of the bound component depends mainly on the

protein (e.g. albumin). The effective diffusion coefficient for albumin in VX2 carcinoma was

measured as 8.89×10-12

m2/s [90]. Since the bound component has a MW of approximately 69


57

kDa [51], its diffusion coefficient in normal tissue estimated by Equation (3-47) is 4.17×10-12

m2/s.

(2) Liposome encapsulated doxorubicin

For diffusion of spherical particles through liquid at low Reynolds number, the diffusion

coefficient in water can be calculated by solving the Einstein-Stokes equation [15].

rTk

D Bw 6

*

(3-48)

where kB is Boltzmann's constant, T* is the absolute temperature, μ is viscosity and r is the radius

of spherical particle. Assuming the liposomes are spherical with a uniform diameter of 100 nm,

their diffusion coefficient at 37oC is 5.82×10

-12 m

2/s.

Yuan found that for liposome particles of 90 nm in diameter, the ratio of permeability in tumour

and diffusion coefficient in water was around 20 m-1

[110], and the permeability was

2.0±1.6×10-10

m/s [26]. Based on these results, the diffusion coefficient is 1.0±0.8×10-11

m2/s.

Based on the Einstein-Stokes equation, the diffusion coefficient is inversely related to the

particle radius. Hence, the diffusion coefficient of liposomes with 100 nm diameter is estimated

to be 9.0±7.2×10-12

m2/s. The diffusion coefficients in tumour and normal tissues adopted in this

thesis are 9.0×10-12

m2/s and 5.8×10

-12 m

2/s, respectively.

Reflection Coefficient (σ)

The reflection coefficient determines the efficiency of the oncotic pressure gradient in driving

transport across the vascular wall. It is related to the sizes of drug and pores on the vasculature

wall [126]. For the same drug, this parameter may vary in different types of tissues [127]. Wolf

et al. [128] measured the reflection coefficient for albumin and found this to be 0.82±0.08. The

sizes of albumin and liposome are 3.5 nm and 100 nm, respectively. The reflection coefficient

for liposome is estimated to be greater than 0.90, hence it is assumed to be 0.95 in this study.

Drug Dose (Dd)

As a common anticancer drug, doxorubicin is widely used in chemotherapy to treat various types

of cancer, such as lymphoma, genitourinary, thyroid, and stomach cancer [3]. By interacting with


58

DNA in cells, doxorubicin can inhibit the process of DNA replication. Because of this

mechanism of action, high concentration of doxorubicin in normal tissues can cause serious

damage to healthy cells, known as side effects. In clinical therapy, the most serious toxicity is

life-threatening cardiomyopathy [129, 130], leading to heart failure. Side effects set a limit to the

lifetime dose a patient can receive, which is approximately 550 mg per unit body surface area [3].

The dose of doxorubicin in clinical use is related to the patient’s body surface area. In each

treatment cycle, the dose is between 50 to 75 mg/m2 [3].

Plasma Pharmacokinetics (Cv)

Doxorubicin concentration in blood plasma is modelled as an exponential decaying function of

time. The form of equations depends on the infusion mode.

For continuous infusion, a tri-exponential decay is assumed based on the plasma

pharmacokinetics of doxorubicin [55, 90].

Tteeα

Aee

α

Aee

α

A

T

DC

Tteα

Ae

α

Ae

α

A

T

DC

tαtαtαTαtαTα

d

dv

tαtαtα

d

dv

222211

321

111

111

3

3

2

2

1

1

3

3

2

2

1

1

(3-49)

where t refers to time, Dd is the dose of doxorubicin and Td is the infusion duration. A1, A2 and A3

are compartment parameters and α1, α2, α3 are compartment clearance rate.

For bolus injection, the terms involving α2, α3, A2 and A3 represent the compartments with much

slower elimination. The drug concentration is assumed to follow an exponential decay based on

the plasma pharmacokinetics of doxorubicin [55, 90].

tα

dv eADC 1

1

(3-50)

Free doxorubicin in plasma can easily bind with proteins, such as albumin. Greene et al. [86]

found that approximately 75±2.7 % doxorubicin is present in the bound form, and the percentage

binding is independent of doxorubicin and albumin concentrations. Hence for direct infusion, the

free (Cfp) and bound (Cbp) doxorubicin concentrations in plasma are given by:


59

(3-51)

where s is the percentage of bound doxorubicin, which is 25% in this study.

3.3. Implementations of Mathematical Models

Usually the location where anticancer agents are administrated into a patient’s body is remote

from the tumour, hence the influence of drug infusion on blood pressure in the tumour is ignored.

In addition, drug transport processes are assumed to have no influence on the interstitial fluid

pressure because of the small size of anticancer agents. In order to generate initial conditions for

the time-dependent numerical simulation, the interstitial fluid flow equations are firstly solved to

obtain a steady-state solution in the entire computational domain. Following this, the obtained

pressure and velocity values are applied at time zero for the simulation of drug transport and

cellular uptake in order to predict the treatment efficacy. The mathematical equations described

above are implemented into ANSYS Fluent, which is described in the following section.

3.3.1. ANSYS Fluent

ANSYS FLUENT is a widely used CFD code based on the finite volume method (FVM). The

physical domain is firstly divided into a large number of computational cells forming a

computational grid on which control volumes are defined where the variable of interest is located

in the centroid. Governing equations in the differential forms are then integrated over each of the

control volumes until the residual between two adjunct iterations is less than a given tolerance.

The conservation principle for the variables in each control volume is expressed by the resulting

discretised equation.

The employed solution algorithm is the SIMPLEC which uses a relationship between velocity

and pressure corrections to enforce mass conservation and to obtain the pressure field. The

momentum, mass and heat transfer equations are discretised using the second order UPWIND

scheme, in which quantities at cell faces are computed using a multi-dimensional linear

reconstruction approach [131]. In this approach, higher-order accuracy is achieved at cell faces

through a Taylor series expansion of the cell-centred solution about the cell centroid. For

vbp

vfp

CsC

sCC

1


60

transient problems, a fully implicit second order backward Euler scheme is adopted, which is

unconditionally stable with respect to time step size. The Gauss-Seidel method is adopted to

update values at nodal points by using the previously computed results as soon as they become

available.

3.3.2. User Defined Routines for Drug Transport

Mass transfer equations describing the transport of drugs are coded in C programming language

by using the User Defined Scalar (known as UDS), which is a function for user defined

programme be dynamically loaded with FLUENT.

These equations are solved based on the interstitial fluid field resolved by solving the continuity

and momentum equations. Drug transport equations are also discretised using the second order

UPWIND scheme, and the tolerance controlling the convergence is set to be 1×10-8

.

3.3.3. User Defined Routines for Heat Transfer

Heat transfer under HIFU heating in blood and tissue is also incorporated via UDS. These

equations are solved with the drug transport equations when simulating thermo-sensitive

liposome-mediated drug delivery. The second order UPWIND scheme is adopted for spatial

discretisation and the convergence tolerance is set as 1×10-10

.

3.3.4. Boundary Conditions

In order to solve the above equations numerically and to ensure physiological relevance of the

solutions, each of the dependent variable equations requires meaningful values at the boundary

of the computational domain. These values are known as boundary conditions. Typical

boundaries involved in the problems dealt with in this thesis include: interface between tumour

regions, tumour-normal tissue interface and external surface of normal tissue. The specification

of boundary conditions is problem-dependent, hence details will be given for each problem

described in individual chapters.

3.4. Summary

The numerical methods and tools described above were applied to a realistic prostate tumour

with a focus on understanding the transport steps of non-encapsulated drugs (Chapter 4), a liver


61

tumour with a focus on elucidating the effect of heterogeneous distribution of blood vessels and

3 prostate tumours with a focus on examining the effect of tumour size and microvascular

density (Chapter 5), and a realistic prostate tumour with a focus on investigating the thermo-

sensitive liposome-mediated drug delivery with hyperthermia induced by high intensity focused

ultrasound (Chapter 6). All computations were carried out on an HP Compaq 8100 Elite CMT

PC.


62

Table 3-1. Parameters for tumour and normal tissues.

Parameter Definition Unit Tumour Tissue Normal Tissue Reference

S/V Surface area of blood vessels per unit tissue volume m-1

20000 7000 [49-51, 88]

Kv Hydraulic conductivity of the micro-vascular wall m/Pa·s 2.10×10-11

2.70×10-12

[49-51, 88]

ρ Density of interstitial fluid kg/m3 1000 1000 [51]

µ Dynamic viscosity of interstitial fluid kg/m·s 0.00078 0.00078 [51]

1/κ Permeability of the interstitial space m-2

4.56×1016

2.21×1017

[49-51, 88]

pv Vascular fluid pressure Pa 2080 2080 [49-51, 88]

πv Osmotic pressure of the plasma Pa 2666 2666 [49-51, 88]

πi Osmotic pressure of interstitial fluid Pa 2000 1333 [49-51, 88]

σT Average osmotic reflection coefficient for plasma proteins 0.82 0.91 [49-51, 88]

KlySly/V Hydraulic conductivity of the lymphatic wall times surface

area of lymphatic vessels per unit volume of tumour tissue

(Pa·s)-1

0 4.17×10-7

[51]

ply Intra-lymphatic pressure Pa 0 0 [51]

Dc Cell density 105cell/m

3 1×10

10 - [90]

VT Total tumour volume m3

5×10-5

3×10-4

[55]

VB Total blood volume in body m3 5×10

-2 5×10

-2 [55]


63

Table 3-2. Parameters for liposome

Parameter Definition Unit in Tumour in Normal Tissue Reference

Pl Liposome permeability of vasculature wall m/s 3.42×10-9

8.50×10-10

[25, 26]

D Liposome diffusion coefficient m2/s 9.0×10

-12 5.8×10

-12 [25, 110]

σl Reflection coefficient for liposome 0.95 1.0 -

CLlp Plasma clearance in tumour

s-1

2.228×10-4

2.228×10-4

[62]


64

Table 3-3. Parameters for doxorubicin.

Parameter Definition Unit Free Doxorubicin Bound Doxorubicin Reference

Ptumour Permeability of vasculature wall in tumour m/s 3.00×10-6

7.80×10-9

[51, 119]

Pnormal Permeability of vasculature wall in normal tissue m/s 3.75×10-7

2.50×10-9

[51, 119]

Dtumour Diffusion coefficient in interstitial fluid of tumour m2/s 3.40×10

-10 8.89×10

-12 [15, 51, 118, 123-125]

Dnormal Diffusion coefficient in interstitial fluid of normal tissue m2/s 1.58×10

-10 4.17×10

-12 [15, 51, 118, 123-125]

σd Osmotic reflection coefficient 0.15 0.82 [51, 128]

ka Doxorubicin-protein binding rate s-1

0.833 - [90]

kd Doxorubicin-protein dissociation rate s-1

- 0.278 [90]

φ Tumour fraction extracellular space 0.4 [90]

Vmax Rate of trans-membrane transport kg/105cells s 4.67×10

-15 - [90, 101]

ke Michaelis constant for transmembrane transport kg/m3

2.19×10-4

- [90, 101]

ki Michaelis constant for transmembrane transport kg/105cells 1.37×10

-12 - [90, 101]

fmax Cell-kill rate constant s-1

1.67×10-5

- [109]

EC50 Drug concentration producing 50% of fmax kg/105cells 5×10

-13 - [109]

A1 parameter for pharmacokinetic model m-3

74.6 74.6 [85, 90]


2.49 2.49 [85, 90]


0.552 0.552 [85, 90]

α1 compartment clearance rate s-1

2.43×10-3

2.43×10-3

[85, 90]

α2 compartment clearance rate s-1

2.83×10-4

2.83×10-4

[85, 90]

α3 compartment clearance rate model s-1

1.18×10-5

1.18×10-5

[85, 90]

kp Cell proliferation rate s-1

3.0×10-6

- [102]

kg Cell physiologic degradation rate s-1

3.0×10-16

- [102]

CLtumour Plasma clearance in tumour s-1

2.43×10-3

0 [85, 114, 132, 133]

CLnormal Plasma clearance in normal tissue s-1

2.43×10-3

0 [85, 114, 132, 133]

4. Non-Encapsulated Drug Delivery to Solid

Tumour

Intravenous administration of drugs in their free form is a traditional drug delivery

method adopted in clinical treatments without specific targeting. Injected drugs

undergo multiple transport processes in systemic blood stream, interstitial space and

tumour cells. The role of drug transport in treatment efficacy is complicated because

of nonlinear interactions between the drug and the microenvironment of tumour and

surrounding normal tissues. An appropriate mathematical model is therefore needed

to gain detailed understanding of the drug transport processes involved and how drug

concentrations and its therapeutic effect may be influenced by infusion modes.

In this chapter, a coupled pharmacokinetics and drug transport model is applied to a

real tumour geometry reconstructed from magnetic resonance image (MRI) to

examine the effects of various administration modes and clinic dose levels. This is

followed by comparisons of drug concentrations and survival fraction of tumour cells

between 2-D and 3-D geometry models.

4. NON-ENCAPSULATED DRUG DELIVERY TO SOLID TUMOUR

66

4.1. Model Description

The mathematic model described in Chapter 3 for free form drug administration is adopted,

which incorporates the key physical and biochemical processes, including time-dependent

plasma clearance, drug transport through the blood and lymphatic vessels, extracellular drug

transport (convection and diffusion), drug binding with proteins, lymphatic drainage, interactions

with the surrounding normal tissue and drug uptake by tumour cells. Anticancer efficacy is

evaluated based on the percentage of survival tumour cells by directly solving the

pharmacodynamics equation using the predicted intracellular drug concentration.

4.1.1. Model Geometry

The geometry of a prostate tumour is reconstructed from images acquired from a patient using a

3.0-Tesla MR scanner (DISCOVERY MR750, GE, Schenectady, New York, USA). Multislice

anatomical images of the prostate were acquired in three orthogonal planes with echo-planer (EP)

sequence, with each image comprising 256 by 256 pixels. Other imaging parameters are given in

Table 4-1.

Table 4-1. MR imaging parameters

Parameter

(unit)

Pixel Size

(mm)

Field of View

(cm)

Slice Thickness

(mm)

Repetition Time

(ms)

Echo Time

(ms)

1.250 32.0 7.000 4000 84.4

An example of the MR images is shown in Figure 4-1 (a) with the tumour region and its

surrounding normal prostatic tissues. Transverse images are processed using image analysis

software Mimics (Materialise HQ, Leuven, Belgium), and the tumour is segmented from its

surrounding normal tissue based on signal intensity values. The resulting smoothed surfaces of

the tumour and normal tissues are imported into ANSYS ICEM CFD to generate computational

mesh for the entire volume.

The tumour shown in Figure 4-1 (b) is located at the corner of normal tissue with a dimension of

47 mm (maximum width) by 38 mm (maximum depth) in 2-D model. The final mesh consists of

64,966 triangular elements which have been tested to produce grid independent solutions.


67

(a)(a)

(b)

Tumour

Normal Tissue

(b)

Figure 4-1. Model geometry: (a) MR image of the prostate tumour (in red) and its surrounding tissue (in pule blue);

(b) the reconstructed 2-D geometry.

4.1.2. Model Parameters

The parameters describing general properties of doxorubicin, tumour and normal tissue have

been given in Tables 3-1 and 3-3 in the preceding chapter. Infusion durations and doses

simulated in this study are described below.

(1) Infusion Duration

Infusion duration is a key parameter in determining the pharmacokinetics of drug in blood.

Previous experiments found that prolonged infusion, such as over 48 hours or 96 hours, did not

contribution to better treatment efficacy [53, 54]. In a numerical study by [55], a 2-hour infusion

was adopted without the consideration of cell proliferation and death as a dynamic process.

Therefore, bolus injection and continuous infusion with infusion duration of 1-hour, 2-hour and

3-hour respectively, are studied.

(2) Infusion Dose

Limited by the serious side effect caused by doxorubicin, the lifetime dose a patient can receive

is approximately 550 mg per unit body surface area. For each treatment cycle, the dose is

between 50 to 75 mg/m2 [3], depending on the body surface area of a patient.

Body surface area (BSA) is determined from the following formula [51]

(4-1) 2

730

m731kg70

kginWeightBSA .

.


68

For a 70 kg patient, dosage of doxorubicin is in the range of 86.5~129.75 mg. Three dose levels,

50, 60 and 75 mg/m2 are examined in this study.

(3) Pharmacokinetics

Doxorubicin concentration in blood plasma is usually described by an exponential function over

time. Better fit to the experimental data was found by using a 3-compartment model.

For bolus injection and continuous infusion in a single administration, the plasma concentration

of doxorubicin can be described by Equation (3-49) and (3-50), respectively. The plasma

concentration in multiple administrations under bolus injection and continuous infusion are given

below by Equation (4-2) and (4-3), respectively [134].

(4-2)

(4-3)

where Dd is the dose of doxorubicin and Td is the infusion duration. A1, A2 and A3 are

compartment parameters and α1, α2, α3 are compartment clearance rate. t* is the time after n

doses (i = 1,2,…,n) given at time tDi.

To account for the high binding rate for doxorubicin with proteins, such as albumin in blood

plasma, and for the fact that the percentage of bound doxorubicin is independent of plasma

concentration of doxorubicin and albumin, the free (Cfp) and bound (Cbp) doxorubicin in blood

are governed by Equation (3-51), where s is the percentage of bound doxorubicin, which is 0.25

in this study [86].

n

i

tt

dviD

*

eADC1

1

1

n

i

dD

*

TttT

TttTTttT

di

di

dD

*

tt

tt

tt

dn

dnn

i

TttT

TttT

TttT

di

di

v

Ttt

eeA

eeA

eeA

T

D

Ttt

eA

eA

eA

T

D

eeA

eeA

eeA

T

D

C

n

diiD*

di

diiD*

didiiD*

i

n

nD*

nD*

nD*

diiD*

di

diiD*

di

diiD*

di

1

3

3

2

2

1

1

3

3

2

2

1

1

1

1

3

3

2

2

1

1

33

2211

3

2

1

33

22

11

1

11

1

1

1

1

1

1


69

4.1.3. Numerical Methods

The mathematical models described above are implemented in ANSYS-Fluent, which is a finite

volume based computational fluid dynamics (CFD) code (ANSYS Inc., Canonsburg, USA). The

momentum and drug transport equations are discretised using the second order UPWIND scheme,

and the SIMPLEC algorithm is employed for pressure-velocity coupling. The Gauss-Seidel

smoothing method is used to update values at nodal points after each iteration step. Convergence

is controlled by setting residual tolerances of the momentum equation and the drug transport

equations to be 1×10-5

and 1×10-8

, respectively.

In order to generate initial conditions for the transient simulation, the interstitial fluid flow

equations are firstly solved to obtain a steady-state solution in the entire computational domain.

Following this, the obtained pressure and velocity values are applied at time zero for the

simulation of drug transport and cellular uptake (shown in Figure 4-2). The second order implicit

backward Euler scheme is used for temporal discretisation, and a fixed time step size of 10

seconds is chosen. This time step is deemed sufficiently fine based on a time step sensitivity test.

The initial doxorubicin concentrations are assumed to be zero in both tumour and the

surrounding normal tissue.

Solving continuity and momentum

equations to obtain steady-state

solution of interstitial fluid field

Solving mass transfer equations to

obtain time-dependent drug

concentration and tumour cell

denisty

IFP IFV

Figure 4-2. Numerical Procedures.

There are two boundary surfaces in this model: an internal boundary between the tumour tissue

and normal tissue, and the outer surface of the normal tissue. At the internal boundary,


70

conditions of continuity of the interstitial pressure and fluid flux are applied. At the outer surface,

a constant relative pressure of 0 Pa and zero flux of drug are assumed.

4.2. Results

The analysis of results obtained from the mathematical model is carried out. The interstitial fluid

field is firstly examined to provide the microenvironment for drug transport and cell killing. This

is followed by comparisons of administration modes and doses under single and multiple

administration treatments in a clinical range. Finally, computational results obtained from the 2-

D and 3-D models are compared.

4.2.1. Interstitial Fluid Field

The interstitial fluid field in tumour and normal tissues plays an important role in drug delivery.

This is because on the one hand, the interstitial fluid velocity and pressure determines the drug

migration in extracellular space by convention; on the other hand, transvascular pressure gradient

influences the drug exchange between microvessels and the interstitial space of tumour. By

solving Equations (3-1)~(3-8) described in Chapter 3 for a vascular pressure of 2080 Pa [49-51,

89], the spatial distribution of interstitial fluid pressure (IFP) is presented in Figure 4-3. There is

a uniform IFP in the tumour and normal tissue. Higher IFP is found in the tumour compared with

its surrounding normal tissue, and there is a large gradient in a thin layer at the interface between

the two regions.

Figure 4- 3. Interstitial fluid pressure in tumour and surrounding normal tissue.

The spatial mean IFP is 1533.88 Pa in the tumour region and 40 Pa in normal tissue. These

results agree with experimental studies which suggested that IFP in tumour was in the range of

586.67 Pa to 4200 Pa [48], and in normal tissue -400 Pa to 800 Pa [135]. This difference in IFP

between tumour and normal tissue can be partially attributed to the special characteristics of

1400

1173

947

720

493

266

40

IFP (Pa)


71

tumour vasculature. On the one hand, vasculature surface is enlarged because blood vessels are

tortuous, elongated, and often dilated in tumour growth. The hydraulic conductivity is increased

by 10-fold as a result of large pores on tumour vasculature wall [49]. This leads to a much easier

fluid exchange across the blood vessel wall. On the other hand, the lack of lymphatic vessels

results in less fluid being drained out of the interstitial space, causing build-up of pressure in

tumour interstitial space. The IFP distribution shown in Figure 4-3 suggests that pressure-

induced convection of interstitial fluid mainly occurs within a thin layer at the interface between

tumour and its surrounding normal tissue.

In theory, high pressure in tumour poses a barrier for drug transport from blood to interstitial

space. However, diffusion driven by concentration gradient and vasculature surface area also

contributes to drug exchange between blood and interstitial space. Previous studies demonstrated

that diffusion was a more important mechanism in determining drug transport across the

vasculature wall because of the ineffective convection owing to the high IFP [51].

4.2.2. Single Administration Mode

In this series of simulations, the full dose of doxorubicin is administrated at the beginning of

each treatment. The influences of two controllable factors, infusion duration and dose level, are

examined over a period of 9 hours following drug administration.

4.2.2.1. Infusion Duration

The cytotoxic effect of doxorubicin on tumour cells is evaluated for bolus injection and

continuous infusions with different infusion durations. Given that doxorubicin is carried by the

blood stream into the tumour and normal tissue, its concentration in blood should be examined

when evaluating its anticancer effectiveness. The time course of doxorubicin blood concentration,

for a total dose of 50 mg/m2 administered by different administration modes, is compared in

Figure 4-4. Because all the drugs are administrated into blood stream in a very short period,

which can be ignored in comparison with the treatment time window, drug concentration in

blood under bolus injection reaches its peak at the very beginning of treatment. Without

continuous supply, the concentration falls rapidly to a low level. Continuous infusion leads to

different results. Since the total dose of drugs are administrated continuously in the infusion

duration, drug concentration in blood increases gradually in this period. At the end of the


72

infusion duration, drug concentration begins to decrease. Therefore, the timing of peak drug

concentration varies according to the infusion duration.

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-3

4.0x10-3

6.0x10-3

8.0x10-3

1.0x10-2

1.2x10-2

Doxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n (

kg/m

3)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

4 5 6 7 8 92.0x10

-5

4.0x10-5

6.0x10-5

8.0x10-5

Figure 4-4. Doxorubicin concentration in plasma as a function of time after start of treatment, for bolus injection and

continuous infusion of various indicated durations (dose = 50 mg/m2).

Comparison of peak values in Figure 4-4 indicates that prolonging the infusion duration results

in low peak doxorubicin blood concentration. This is because doxorubicin is being continuously

cleared up after administration. Since high doxorubicin concentration in blood is associated with

increased risk of cardiotoxicity, bolus injection may not be a preferred option while continuous

infusion with longer duration may help reduce this risk. Comparison also shows that although

fast infusion leads to higher concentrations at the beginning of the treatment, doxorubicin

concentration in blood falls much faster after the infusion ends.

Free and bound doxorubicin extracellular concentrations in tumour and its surrounding normal

tissue are shown in Figure 4-5 and 4-6, respectively. Regardless of the injection mode, both free

and bound doxorubicin concentrations increase rapidly during the initial period following drug

administration. Doxorubicin concentrations in tumour interstitial space increase faster and reach

a much higher peak with bolus injection. The peak value for continuous infusion decreases with

the increase in infusion duration. Although doxorubicin concentration drops to a low level after

infusions end for all modes of administration, continuous infusions are able to maintain a slightly

higher concentration than bolus injection. Comparing doxorubicin extracellular concentrations in

Figures 4-5 and 4-6 with the concentration in blood in Figure 4-4, the concentration curves have

identical shapes for a given administration mode. This means that drug concentration in blood

has a direct influence on the extracellular concentration for both free and bound doxorubicin.


73

0 1 2 3 4 5 6 7 8 9

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

4 5 6 7 8 90.0

8.0x10-6

1.6x10-5

2.4x10-5

3.2x10-5(a)

0 1 2 3 4 5 6 7 8 9

0.0

1.0x10-3

2.0x10-3

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

7.0x10-3

8.0x10-3

9.0x10-3

Boudn D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

4 5 6 7 8 90.0

3.0x10-5

6.0x10-5

9.0x10-5

1.2x10-4

(b)

Figure 4-5. Spatial mean (a) free and (b) bound doxorubicin extracellular concentration in tumour as a function of

time under bolus injection and different infusion durations (dose = 50 mg/m2).

Because binding with protein is a dynamics process, both free and bound doxorubicin

concentrations vary with time. Comparing Figure 4-5 (a) and (b), it is clear that the time course

of free and bound concentrations shares a similar trend, but bound doxorubicin concentration is

approximately 3 times of the free doxorubicin concentration, indicating that most doxorubicin in

the interstitial fluid is in bound form. Since only free form doxorubicin can be taken up by

tumour cells [55, 90], the high binding rate is expected to reduce the intracellular concentration

of doxorubicin.

0 1 2 3 4 5 6 7 8 9

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

(a)

0 1 2 3 4 5 6 7 8 9

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

(b)

Figure 4-6. Spatial mean (a) free and (b) bound doxorubicin extracellular concentration in normal tissue as a

function of time under bolus injection and different infusion durations. (dose = 50 mg/m2).

Figure 4-7 shows the spatial distribution of free doxorubicin extracellular concentration in

tumour and its surrounding normal tissue at 0.5 hour under 2-hour continuous infusion. There is

a difference in drug concentration between tumour and surrounding normal tissue. Since the


74

properties of doxorubicin and tissues are assumed to be uniform in each region, the

corresponding distribution is also uniform except near the tumour boundary where a large

concentration gradient exists. This thin layer of steep concentration gradient represents the

exchange of drug between tumour and normal tissue.

3.58×10-5

3.38×10-5

3.19×10-5

3.00×10-5

2.79×10-5

2.60×10-5

2.40×10-5

Cf (kg/m3)

Figure 4-7. Spatial distribution of extracellular concentration of free doxorubicin in tumour and normal tissues (2-

hour infusion, time = 0.5 hour).

As has already been shown in Figure 4-3, the interstitial fluid pressure in tumour is much higher

than in normal tissue; this pressure difference drives interstitial fluid flow from the tumour to

normal tissue, and doxorubicin is also carried by this flux. Moreover, doxorubicin diffuses from

tumour where its concentration is high compared to the surrounding tissue because of the large

concentration gradient in the interstitial space. The transport from tumour to normal tissue needs

to be minimised in order to maintain a high level of doxorubicin concentration in tumour.

0 1 2 3 4 5 6 7 8 9

0.0

0.5

1.0

1.5

2.0

Doxoru

bic

in I

ntr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(ng/1

05ce

lls)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

Figure 4-8. Doxorubicin intracellular concentration as a function of time, for bolus injection and continuous

infusions of various durations (dose = 50 mg/m2).

Figure 4-8 presents the intracellular doxorubicin concentration in tumour under bolus injection

and continuous infusions with various infusion durations. The intracellular concentration


75

displays a sharp rise at the beginning until it reaches a peak, and then decreases. During the

initial phase, the rate of increase in doxorubicin concentration slows down with increase of

infusion duration. Compared to continuous infusions, bolus injection leads to a higher peak and a

faster reduction in intracellular concentration which also remains at a much lower level after

cessation of drug administration.

For continuous infusions, longer infusion durations tend to slow down the increase in

intracellular concentration at the initial phase of treatment, and produce a lower peak. Here, the

highest intracellular concentration is found for a 1-hour continuous infusion. Moreover, for

continuous infusion with different durations, intracellular concentrations reach a very similar

level after drug infusions terminate.

0 1 2 3 4 5 6 7 8 9

76

80

84

88

92

96

100

Fra

ctio

n o

f S

urv

ival

Cel

l (%

)

Time (hr)

bolus injection

1-hour infusion

2-hour infusion

3-hour infusion

Figure 4-9. Predicated percentage tumour cell survival under bolus injection and continuous infusions of various

durations (dose = 50 mg/m2).

Figure 4-9 shows the percentage of survival tumour cells by applying the pharmacodynamics

model described by Equation (3-30). As can be observed, the cytotoxic effect of bolus injection

on tumour cells is significantly lower than that of continuous infusions. This is because of the

rapid clearance of doxorubicin from blood (shown in Figure 4-4), resulting in a dramatic

reduction in extracellular drug concentration which is reduced to approximately zero after 5~6

hours (shown in Figure 4-5). Without enough doxorubicin accumulating in tumour cells, the cell

killing rate and physical degradation rate become slower than cell proliferation rate, and hence

tumour cell density begins to increase after reaching the minimum value at around 5 hours after

the cessation of drug administration. On the contrary, the extracellular drug concentration with

continuous infusion stays at a low but non-zero level for a longer period to supply sufficient


76

doxorubicin for cell killing. This is the reason for the continuous reduction in survival fraction of

tumour cells with continuous infusion in the simulation time window.

However, the effect of infusion duration seems to be insignificant for the limited range of

infusion durations examined, with a difference of up to 2% between the slower (3 hours) and

faster infusion (1 hour). The best therapeutic efficacy is not achieved with the infusion duration

that leads to the highest intracellular concentration. Results show that 1-hour infusion leads to

high cell killing rate at the beginning of treatment because of high intracellular concentration

(shown in Figure 4-8). However, the killing rate slows down due to a rapid reduction in

intracellular concentration. This low level of intracellular concentration sustains to the end of the

simulation time window, and the therapeutic efficacy is the worst among the continuous infusion

modes examined. On the contrary, although a 3-hour infusion results in a slower increase and

lower peak, as shown in Figure 4-8, the intracellular concentration is at a higher level after peak

time and the cytotoxic effect is better than others. This indicates that the final treatment outcome

is more dependent on the intracellular concentration over the entire exposure time rather than its

peak value.

4.2.2.2. Effect of Dose Level

Another controllable parameter in chemotherapy is the dose level of anticancer drugs. In current

clinical use, doxorubicin dose ranges from 50 mg/m2 to 75 mg/m

2 for a 70kg patient, hence the

effect of dose on drug effectiveness is examined within this range.

Comparison is made between different doses (50, 60 and 75 mg/m2) under a 2-hour continuous

infusion, simulating treatment doses used for a 70 kg patient. Blood concentration of free

doxorubicin presented in Figure 4-10 (a) shows that increasing the dose results in linear elevation

of drug concentration. Since following the same pharmacokinetics, blood concentration of bound

doxorubicin is believed to share the same feature. As shown in Figure 4-10 (b) and (c),

extracellular concentration of doxorubicin in both free and bound forms are increased because of

the close relationship between concentration in blood and in interstitial space. As a consequence,

more drugs accumulate in tumour cells under a high dose administration.

Results shown in Figure 4-10 (d) show that intracellular drug concentration is not proportional to

the administrated dose level. In summary, doxorubicin concentrations in all regions increase with


77

the dose level, since a high dose increases the amount of drug delivered to both the tumour and

normal tissues. In addition, the high concentration in blood may also increase the risk of heart

failure.

0 1 2 3 4 5 6 7 8 9

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

Doxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n (

kg/m

3)

Time (hr)

75 mg/m2

60 mg/m2

50 mg/m2

(a)

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

1.4x10-4

1.6x10-4

1.8x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

75 mg/m2

60 mg/m2

50 mg/m2

(b)

0 1 2 3 4 5 6 7 8 9

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

75 mg/m2

60 mg/m2

50 mg/m2

(c)

0 1 2 3 4 5 6 7 8 9

0.0

0.5

1.0

1.5

2.0

2.5D

oxoru

bic

in I

ntr

acel

lula

r C

once

ntr

atio

n

(ng/1

05ce

lls)

Time (hr)

75 mg/m2

60 mg/m2

50 mg/m2

(d)

Figure 4-10. Spatial mean doxorubicin concentration in tumour as a function of time under 2-hour infusion

duration.(a) doxorubicin blood concentration, (b) free doxorubicin extracellular concentration, (c) bound

doxorubicin extracellular concentration, and (d) intracellular concentration.

The percentage of survival tumour cells is compared in Figure 4-11 by solving the

pharmacodynamics equation based on the predicted intracellular concentration. The results show

that the cytotoxic effect of doxorubicin on tumour cells increases with the dose level, with the

difference between low (50 mg/m2) and high doses (75 mg/m

2) being 2.5% at 9-hour following

drug administration. Compared to the administrated dose level, this improvement in treatment

outcome is not obvious.


78

0 1 2 3 4 5 6 7 8 9

75

80

85

90

95

100

Fra

ctio

n o

f S

urv

ival

Cel

l (%

)

Time (hr)

75 mg/m2

60 mg/m2

50 mg/m2

Figure 4-11. Predicated percentage of tumour cell survival under various doses.

The predicated free and bound doxorubicin concentration in the interstitial space of normal tissue

with different drug dosages is compared in Figure 4-12. Results show that increasing the dose

results in elevation of extracellular concentration of doxorubicin in both free and bound forms.

This indicates that the risk of cell killing in normal tissue may also be increased at higher dose.

0 1 2 3 4 5 6 7 8 9

0.0

4.0x10-5

8.0x10-5

1.2x10-4

1.6x10-4

2.0x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

75 mg/m2

60 mg/m2

50 mg/m2

(a)

0 1 2 3 4 5 6 7 8 9

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

75 mg/m2

60 mg/m2

50 mg/m2

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(b)

Figure 4-12. Spatial mean doxorubicin concentration in normal tissue as a function of time under 2-hour infusion

duration (for a 70 kg patient). (a) free doxorubicin extracellular concentration, (b) bound doxorubicin extracellular

concentration.

4.2.3. Multiple Administration Mode

Numerical studies are carried out under continuous infusion and bolus injection with multiple

administrations for 2, 4 and 8 times within the first 48 hours of treatments with intervals of 24,

12 and 6 hours, respectively. The infusion duration is set as 2-hour for each of continuous

infusion, and the simulation is performed for 96 hours.


79

4.2.3.1. Continuous Infusion

Figure 4-13 shows the predicted time course of plasma doxorubicin concentration, for a total

dose of 50 mg/m2 administrated by single and multiple 2-hour continuous infusions. Regardless

of the administration mode, the drug concentration in blood reaches a peak after the end of each

administration. Results show that multiple-infusion can significantly reduce the peak plasma

concentration, and this reduction effect increases with the number of infusion. Administrating the

same dose of doxorubicin 8-times gives a 20% lower peak concentration compared with a single

administration. Hence multiple-infusion is likely to reduce the risk of heart electrophysiology

dysfunction and muscle damage caused by doxorubicin. However, quantitative comparison of

blood concentration after administration shows that multiple-infusion can also lead to a slightly

high level of drug concentration in plasma over time.

Figure 4-13. Doxorubicin intravascular concentration under single and multiple continuous infusions as a function of

time (total dose = 50 mg/m2).

Figure 4-14 shows the time course of free and bound doxorubicin concentration in the interstitial

fluid for different infusion modes. During each infusion period, all forms of doxorubicin begin to

accumulate in the interstitial space since doxorubicin is continuously administrated into the

circulation system. Drug concentration reaches a peak value at the end of each infusion, and then

falls to a lower level as time proceeds. Results show that free and bound doxorubicin

concentrations share the same trend for a specific administration mode. Multiple-infusion can

effectively reduce the peak extracellular concentration and lead to a slightly higher concentration

level over time.

0 8 16 24 32 40 48 56 64 72 80 88 96

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

Doxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

(kg/m

3)

Time (hr)

single_infusion

twice_infusion

4 times_infusion

8 times_infusion


80

0 12 24 36 48 60 72 84 96

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

single_infusion

twice_infusion

4_times_infusion

8_times_infusion

(a)

0 12 24 36 48 60 72 84 96

0.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

single_infusion

twice_infusion

4_times_infusion

8_times_infusion

(b)

Figure 4-14. Spatial mean doxorubicin extracellular concentration as a function of time under single and multiple

continuous infusion. (a) free doxorubicin, and (b) bound doxorubicin (total dose = 50 mg/m2).

Intracellular doxorubicin concentrations in tumour with signal and multiple-infusion modes are

compared in Figure 4-15. It is clear that single infusion gives the highest peak intracellular

concentration, while multiple-continuous-infusions have lower but multiple peaks which help to

keep a sustained level of intracellular concentration over the entire treatment period.

Figure 4-15. Time course of doxorubicin intracellular concentration under single and multiple continuous infusions.

(total dose = 50 mg/m2)

Figure 4-16 shows the survival fraction of tumour cells under various administration modes by

applying the pharmacodynamics model described in Equation (3-30). As can be observed, single

infusion is more effective at cell killing in the first 34 hours but less effective than multiple-

infusions afterwards. Comparing different fractionations under the multiple-infusion mode, cell

killing rate is more steady and the fraction of survival tumour cells reaches a lower level as the

0 12 24 36 48 60 72 84 96

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Intr

acel

lula

r C

once

ntr

atio

n i

n T

um

our

(ng/1

05ce

lls)

Time (hr)

single_infusion

twice_infusion

4_times_infusion

8_times_infusion


81

number of infusions increases. Results show that the 8-times continuous infusion can improve

the anticancer effectiveness by 7% when compared with a single infusion.

Figure 4-16. Time course of tumour survival fraction under single and multiple continuous infusions.

(dose = 50 mg/m2)

Tumour cell density is determined by the rate of cell killing, cell proliferation and degradation

rates, hence the survival fraction may increase when intracellular concentration falls below a

threshold value during administration intervals. By increasing administration times while

keeping the total administration dose the same, multiple-infusion mode is effective at increasing

doxorubicin concentrations between administration cycles (shown in Figure 4-13, 4-14 and 4-15).

Therefore, the rate of cell death (including cell killing and degradation) is faster than that of cell

proliferation, and as a consequence the fraction of survival tumour cells continuous to fall for a

longer period under multiple-infusion treatments than under a single administration.

Taking results shown in Figures 4-13, 4-14 and 4-15 together, it becomes clear that keeping

doxorubicin at a relatively higher level for a longer period leads to a better anticancer

effectiveness. The treatment outcome is related but not solely determined by the peak extra- and

intra-cellular concentrations. This is because the time-scales of drug transport processes from

blood to the interior of tumour cells are different. Although some of these time scales are

determined by the properties of anticancer drug and tissues, the rate-limiting step is crossing the

cell membrane [55]. The slow cellular influx requires the concentration to sustain at a high level

for a sufficient period of time in order to let more doxorubicin accumulate in the cells.

0 12 24 36 48 60 72 84 96

60

65

70

75

80

85

90

95

100

Surv

ival

Fra

ctio

n (

%)

Time (hr)

single_infusion

twice_infusion

4_times_infusion

8_times_infusion


82

Time course profiles of free and bound doxorubicin concentration in the interstitial space of

normal tissue are presented in Figure 4-17. Results indicate that multiple-continuous-infusion

lead to lower peak doxorubicin concentration but a slightly higher concentration level after

administration ends. Compared with doxorubicin intravascular concentration shown in Figure 4-

13, the accumulation of doxorubicin in normal tissue is found to be strongly dependent on

doxorubicin concentration in blood.

0 8 16 24 32 40 48 56 64 72 80 88 96

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

single_infusion

twice_infusion

4 times_infusion

8 times_infusion

(a)

0 12 24 36 48 60 72 84 96

0.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

single_infusion

2 times_infusion

4 times_infusion

8 times_infusion

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(b)

Figure 4-17. Spatial mean doxorubicin extracellular concentration in normal tissue as a function of time under single

and multiple continuous infusion against time. (a) free doxorubicin, (b) bound doxorubicin (total dose = 50 mg/m2).

Figure 4-18 compares the single infusion with a high dose and multiple-infusions with a low

dose. In Figure 4-18 (a) and (c), intracellular concentration resulted by multiple-infusions with

low dose is obviously low, and quantitative comparisons show that 8 times infusion with half

dose can reduce the peak concentration by an order of magnitude

The survival fractions of tumour cells under different administration modes are shown in Figure

4-18 (b) and (d). Numerical results show that differences in the maximum cytotoxic effect

between high drug dose single infusion and low drug dose 8 times infusion are 8% with a 50%

dose and 3% with a 67% dose for multiple infusions. In addition, the recovery of tumour cells is

slower with low dose 8 times infusion, as shown in Figure 4-18 (d). Consequently, multiple-

continuous-infusion seems to offer improved treatment outcome overall.


83

Figure 4-18. Comparison of various doses under single and multiple-infusions. (a) intracellular concentration and (b)

cell survival fraction under single infusion with 50 mg/m2 and multiple infusions with 25 mg/m

2 as a function of

time ; (c) intracellular concentration and (d) cell survival fraction under single infusion with 75 mg/m2 and multiple

infusions with 50 mg/m2 as a function of time.

4.2.3.2. Bolus Injection

The doxorubicin concentration in plasma under various bolus injection schedules are shown in

Figure 4-19 (a). Since the injection duration is short enough to be ignored, peak blood

concentration is reached at the beginning of each administration. Fast plasma clearance is the

reason for the rapid fall of blood concentration after administration. Quantitative comparisons

show that the peak intravascular concentration is inversely related to the number of injections.

This means that multiple bolus injection can reduce the peak concentration in plasma, and hence

may help reduce the risk of cardiotoxicity.

The free and bound doxorubicin concentration in the interstitial space is shown in Figure 4-19 (b)

and (c). Because doxorubicin needs to permeate through the vasculature wall and then enter the

0 12 24 36 48 60 72 84 96

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

50 mg/m2_single_infusion

25 mg/m2_twice_infusion

25 mg/m2_4_times_infusion


Intr

acel

lula

r C

once

ntr

atin

in T

um

our

(ng/1

05ce

lls)

Time (hr)

(a)

0 12 24 36 48 60 72 84 96

65

70

75

80

85

90

95

100





Surv

ival

Fra

ctio

n i

n T

um

our

(%)

Time (hr)

(b)

0 8 16 24 32 40 48 56 64 72 80 88 96

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Intr

acel

lula

r C

once

ntr

atio

n i

n T

um

our

(ng/1

05ce

lls)

Time (hr)





(c)

0 8 16 24 32 40 48 56 64 72 80 88 96

55

60

65

70

75

80

85

90

95

100

Surv

ival

Fra

ctio

n i

n T

um

our

(%)

Time (hr)





(d)


84

interstitial space, its extracellular concentration shares a similar pattern to the intravascular

concentration shown in Figure 4-19 (a).

Figure 4-19. Comparison of doxorubicin concentration under single and multiple bolus injections as a function of

time (total dose = 50 mg/m2). (a) intravascular concentration, (b) free doxorubicin extracellular concentration, (c)

bound doxorubicin extracellular concentration, and (d) intracellular concentration

The predicted time courses of doxorubicin intracellular concentration, for a total dose of 50

mg/m2 administered by different bolus injection schedules, are compared in Figure 4-19 (d).

Results show that single injection leads to the highest peak intracellular concentration, and the

peak value decreases with the number of injections. Although sharing a similar trend to

intravascular concentration, intracellular concentration takes longer to drop to its lowest level

which is approximately zero. This is because the time-scale for crossing cell membrane is much

longer than that of plasma clearance [55].

Figure 4-20 presents the survival fraction of tumour cell based on the predicted intracellular

concentration over time. The cytotoxic effect of single bolus injection on tumour cells is

0 4 8 12 16 20 24 28 32 36 40 44 48

0.000

0.002

0.004

0.006

0.008

0.010

0.012

single_injection

twice_injection

4 times_injection

8 times_injection

Doxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

(kg/m

3)

Time (hr)

(a)

0 6 12 18 24 30 36 42 48

0.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

single_injection

twice_injection

4 times_injection

8 times_injection

(b)

0 6 12 18 24 30 36 42 48

0.0

1.0x10-3

2.0x10-3

3.0x10-3

4.0x10-3

5.0x10-3

single_injection

twice_injection

4 times_injection

8 times_injection

Bound D

oxourb

icin

Extr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

(c)

0 6 12 18 24 30 36 42 48

0.0

0.5

1.0

1.5

2.0

2.5

single_injection

twice_injection

4 times_injection

8 times_injection

Intr

acel

lula

r C

once

ntr

atio

n i

n T

um

our

(ng/1

05ce

lls)

Time (hr)

(d)


85

significantly lower than that of multiple injections, and the difference is up to 15% between the

single and 8-times injections.

Figure 4-20. Tumour survival fraction as a function of time with single and multiple bolus injection


Free and bound doxorubicin concentration in normal tissue under different bolus injection

schedules are compared in Figure 4-21. Results show that peak concentration of doxorubicin in

normal tissue can be reduced by increasing the number of injections.

Figure 4-21. Comparison of doxorubicin concentration under single and multiple bolus injections as a function of

time (total dose = 50 mg/m2). (a) free and (b) bound doxorubicin concentration in normal tissue.

The treatment outcomes under 8-times bolus injection and a single 2-hour continuous infusion

for the total dose of 50 mg/m2 non-encapsulated doxorubicin are compared in Figure 4-22.

Although multiple injections help to improve the anticancer effectiveness as already shown in

0 8 16 24 32 40 48 56 64 72 80 88 96

72

76

80

84

88

92

96

100

single

twice

4 times

8 timesSurv

ival

Fra

ctio

n o

f T

um

our

Cel

ls (

%)

Time (hr)

0 6 12 18 24 30 36 42 48

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

single_injection

twice_injection

4 times_injection

8 times_injection

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(a)

0 6 12 18 24 30 36 42 48

0.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

1.6x10-3

single_injection

twice_injection

4 times_injection

8 times_injection

Bound D

oxourb

icin

Extr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(b)


86

Figure 4-20, results obtained with 8-times bolus injection is still not as good as that with 2-hour

continuous infusion for the same dose.

Figure 4-22. Comparison of survival fraction between single continuous infusion and 8 times bolus injection.


4.2.4. Comparison of 2-D and 3-D Models

All the simulations presented so far are based on a 2-D model reconstructed from a selected MR

image of a transverse section. However, a real tumour and its surrounding normal tissues are

three-dimensional (3-D). Hence, it is necessary to evaluate the need for 3D simulation by

quantitative comparison of results based on 2-D and 3-D models.

Figure 4-23. 3-D model geometry.

For this purpose, a 3-D model is reconstructed from the same series of MR images acquired from

a patient with prostate tumour, and a representative MR image is already shown in Figure 4-1 (a).

The reconstructed 3-D geometry is shown in Figure 4-23, where the tumour region is highlighted

0 8 16 24 32 40 48 56 64 72 80 88 96

65

70

75

80

85

90

95

100

Fra

ctio

n o

f S

urv

ival

Tum

our

Cel

ls (

%)

Time (hr)

8-times bolus injection

2-hour continuous infusion


87

in orange which is surrounded by normal prostatic tissues in pale blue. The dimension of the

reconstructed 3-D model is approximately 40 mm in length, and the tumour and normal tissue

volumes are 2.28×10-6

m3 and 4.42×10

-5 m

3, respectively. The final mesh consists of 757,453

tetrahedral elements which have been tested to produce grid independent solutions.

Numerical simulation has been carried out for a 2-hour continuous infusion of 50 mg/m2

doxorubicin. Comparisons of drug concentration and drug cytotoxic effect are summarized in

Figure 4-24. The time courses of free doxorubicin extracellular concentration based on the 2-D

and 3-D models are compared in Figure 4-24 (a). No difference can be observed between the two

models. Quantitative differences in drug concentrations and tumour cell density are summarized

in Figure 4-24 (b), which shows that differences between the 2-D and 3-D models are less than

0.25%, justifying the use of 2-D models in this study. These results agree with the work of other

researchers [79]. However, it is important to point out that the high level of consistence between

the 2-D and 3-D models is mainly due to the fact that both drug and tissue properties are

assumed to be uniform in each region. The effect of heterogeneous distribution of

microvasculature in tumour will be addressed in Chapter 5.

Figure 4-24. Comparison of drug concentration and cell density between 2-D and 3-D models. (a) free doxorubicin

extracellular concentration in tumour as a function of time, (b) maximum percentage differences in drug

concentration and tumour cell density (total dose = 50 mg/m2)

4.3. Discussion

After intravenous infusion, anticancer drugs need to experience the following steps before

reaching the interior of tumour cells: clearance from plasma, passing through the

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

2-D Model

3-D Model

(a)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Dc

Ci_tumour

Cfe_normal_tissue

Cbe_tumour

Cfe_normal_tissue

Per

centa

ge

of

Dif

fere

nce

(%

)

Cfe_tumour

(b)


88

microvasculature wall, diffusion and convection in extracellular space and crossing tumour cell

membrane. The time scale for each of these transport steps is different owing to the different

properties of microvasculature, interstitial fluid and the kinetics of cellular uptake. Data in the

literature suggest that there is a delay between intracellular and extracellular drug concentration

levels [55]. This is the reason why extracellular concentration needs to be maintained at a high

level for long enough to enable cellular uptake. Simulation results in this study show that peak

intracellular doxorubicin concentration is achieved 0.8~19 minutes later than the corresponding

peak extracellular concentration. Comparisons between single and multiple administration

methods as well as dose levels suggest that a more effective administration is more successful in

maintaining high levels of extracellular doxorubicin over longer time (shown in Figure 4-5, and

4-10 (b,c)), and thereby raising intracellular levels (shown in Figure 4-8, and 4-10 (d)) and

enhancing cellular uptake.

Anticancer effectiveness is usually evaluated by cell survival fraction, which can be predicted by

different mathematical models. As an improvement of the AUC model [104], the peak

intracellular concentration model [55, 90] was found to have better agreement to experimental

data. However, this model predicts the best cell kill fraction for the entire treatment only, without

giving detailed information on the dynamic variation of tumour cell density over time.

Knowledge of temporal variation of cell survival is important since both extra- and intra-cellular

drug concentrations change with time during a treatment period. The pharmacodynamics model

employed in this study overcomes this shortcoming by providing temporal profiles of cell

survival fraction. In addition, the model allows cell proliferation and degradation to be included,

making it a step closer to mimicking an in vivo environment. With the consideration of cell

proliferation and degradation in the present study, results shown in Figures 4-8, 4-10, 4-14, 4-15

and 4-16 demonstrate that sustained levels of extracellular concentration lead to higher

intracellular concentration over time, which can cause more effective tumour cell killing.

Comparisons of drug concentration and survival fraction of tumour cells between 2-D and 3-D

models show that the difference between the two models is negligible. However, this verdict is

based on a number of assumptions. Firstly, the properties of tumour and normal tissues are

assumed to be constant. These may vary with the tissue type, growth stage and individual

conditions of a patient. Secondly, transport properties and vascular distribution are assumed to be


89

uniform, which is not true in a real tumour. All these properties will affect the way anticancer

drugs are transported to tumour cells, and eventually the effectiveness of anticancer treatment.

The model parameters adopted in this study correspond to average and representative values for

tumour and normal tissues extracted from the literature, which may be sufficient for qualitative

study, but more information would be required for detailed quantitative analysis.

4.4. Summary

In this chapter, various administration modes, schedules and dose levels are compared based on a

2-D prostate tumour model with a realistic geometry. The results show that better anticancer

effectiveness is achieved with administration methods that are able to maintain a high level of

extracellular concentration over the entire treatment period.

Results on single administrations show that bolus injection is much less effective in tumour cell

killing than continuous infusion, the latter being not only effective in improving the cytotoxic

effect of doxorubicin on tumour cells, but also in reducing peak level of doxorubicin in blood

stream. Modelling results also show that rapid continuous infusion leads to faster cell killing at

the beginning of the treatment, but slow infusion modes perform better over time.

Increasing drug doses can increase concentration levels in all regions of tumour, and improve the

effectiveness of anticancer treatments. Whilst consideration must be given to potential side

effects, which limit the dose used in clinical treatment, the highest tolerable dose gives the best

result.

Multiple-administration can improve the efficacy of anticancer treatments, while significantly

reducing the peak intravascular concentration and therefore lower the risk of cardiotoxicity

caused by doxorubicin in blood stream. Treatment outcomes may be improved by increasing the

number of administrations. The simulation results indicate that reducing the total dose while

increasing the number of continuous infusions is an effective strategy to reduce plasma drug

concentration, hence the associated cardiotoxicity, although at the expense of a slight reduction

in predicted anticancer effectiveness.

5. Effect of Tumour Properties on Drug

Delivery

The properties of tumours are decisive factors in drug transport and final treatment

efficacy. Since in most chemotherapies anticancer drugs are carried via the systemic

circulation, the characteristics of tumour vasculature plays a crucial role in drug

delivery. On the one hand, vasculature in malignant tumours is highly abnormal and

heterogeneous. A typical tumour consists of a necrotic core with no functional blood

vessels in the centre, semi-necrotic regions with less capillaries and veins, and a

tumour advance front which is rich in arterioles, capillaries and venules [49]. On the

other hand, tumour vasculature density varies in the growth process, with less

vasculature in large, advanced tumours.

In this chapter, a liver tumour model incorporating details of microvascular

distribution is adopted first to examine the effect of heterogeneous vasculature

distribution on drug delivery. This is followed by studies of drug transport in prostate

tumours with different sizes and vasculature density.

5. EFFECT OF TUMOUR PROPERTIES ON DRUG DELIVERY

91

5.1. Heterogeneous Vasculature Distribution

The effect of vasculature distribution is investigated in a liver tumour with anticancer drugs

being directly administrated through intravenous infusion. The drug transport processes and their

governing equations are consistent with those used in Chapter 4. The unique tumour geometry

and parameters are described below.


The geometry of a liver tumour is reconstructed from the post contrast MR images acquired from

a patient using a 3.0-Tesla MR scanner (DISCOVERY MR750, GE, Schenectady, New York,

USA). Multislice anatomical images of the liver were acquired in three orthogonal planes with

GR sequence at three time points after systemic infusion of the tracer Gd-DTPA (gadopentetate

dimeglumine, MW∼0.57 kDa), with each image comprising 256×256 pixels. Other imaging

parameters are given in Table 5-1.


Parameter

(unit)

Pixel Size

(mm)

Field of View

(mm)

Slice Thickness

(mm)

Repetition Time

(ms)

Echo Time

(ms)

1.3 420 5.0 3.808 1.808

In order to build a 2D model of the tumour and its surrounding normal tissue, a representative

transverse image is chosen, as shown in Figure 5-1 (a). The chosen section covers the maximum

dimension of the tumour. Image processing is carried out by using Mimics (Materialise HQ,

Leuven, Belgium), with the tumour and its sub-regions being segmented based on pixel

intensities and contrast enhancement. It can be seen that the tumour is located at the corner with

a maximum dimension of 41 mm and an area of 9.0×10-4

m2. The tumour is divided into 6 sub-

regions based on differences in signal intensities which are correlated to vessel densities as

described later. Region_1 is further divided into two regions in order to compensate for any

potential boundary effect caused by the lack of surrounding normal tissue. The resulting

smoothed contours of the tumour and normal tissues are imported into ANSYS ICEM CFD to

generate a computational mesh. The dimensions of the reconstructed model are approximately

118.8 mm (maximum width) by 126.8 mm (maximum depth), and the area of the normal tissue is

8.7×10-3

m2. The final mesh consists of 88,594 triangular elements. This is obtained based on


92

mesh sensitivity tests which are carried out until the difference in predicted drug concentration

between the adopted mesh and a 10-time finer mesh is less than 1%.

(a)

Normal Tissue

Region_3

Region_5

Region_1_1

Region_4

Region_2

Region_6

Region_1_2

(b)

Figure 5-1. Model geometry: (a) MR image of the liver tumour (in red) and its surrounding tissue (in pule blue); (b)

the reconstructed 2-D geometry.


A large number of model parameters are required for the computational simulation; these include

physical and transport properties for doxorubicin, tumour and normal tissues. Essential model

parameters adopted in the present study are summarized in Tables 3-1 and 3-3. All the geometric

and transport parameters are considered to be independent of time, since the growth of tumour

and its surrounding normal tissue is negligible during one simulated cycle.

The parameter used to describe the density of vasculature is the surface area of blood vessels per

unit volume of tumour tissue (S/V). Since Gd-DTPA is an approved contrast agent for imaging of

blood vessels and inflamed or diseased tissue with leaky blood vessels, this MRI-visible tracer is

first systemically administrated before MR imaging. The concentration of Gd-DTPA tracer in the

interstitial fluid can be described by

GdpvGdGdpGd CFCC

V

SP

dt

dC__ 1 (5-1)

where t is time, P is the permeability of the tracer, S/V is the surface area of blood vessels per

unit volume of tumour tissue, CGd and Cp represent the tracer concentration in the interstitial


93

fluid and blood, respectively, Fv is the filtration rate from the blood vessels per unit volume of

tumour tissue, and σ is the osmotic reflection coefficient of the tracer (σ = 0.6 [136])

The average interstitial concentration of Gd-DTPA has been found to be proportional to the

signal relative intensity enhancement [136]:

1100_

0_

rTS

SSC

Gd

GdGd

Gd

(5-2)

where SGd and SGd_0 represent the signal with and without the tracer, Gd-DTPA. T10 denotes the

relaxation time and r1 is the relaxivity, both of which are constant and can be cancelled out

during the evaluation of S/V.

The spatial-mean signal intensity of each tumour region at three different time points is

calculated, and the relative value of surface area of blood vessel per tumour size, (S/V)*, can be

obtained by interpolation. The ratio of (S/V)* in each tumour region to the average value for the

entire tumour region is then used to calculate values for S/V, by scaling with the standard value

from the literature which is 200 cm-1

. The scaled values are summarized in Table 5-2 in

descending order of microvasculature density.

Table 5-2. The relative and scaled values of surface area of blood vessel per tumour size (cm-1

)

Average Region_1 Region_2 Region_3 Region_4 Region_5 Region_6

(S/V)* 776.52 1741.73 860.84 554.73 502.23 428.69 324.48

S/V 200.00**

448.60 221.72 142.88 129.35 110.41 83.57

** baseline value from literature [49-51]

5.1.3. Numerical Methods

The mathematical equations are solved by means of a finite volume based computational fluid

dynamics (CFD) code, ANSYS FLUENT, together with tailor-made routines using the User

Defined Scalar (UDS) function. The second order UPWIND scheme is adopted for spatial

discretization, while temporal discretization is achieved by employing the second order implicit

backward Euler scheme. Regarding boundary conditions, the external boundaries are the outer

surface of the normal tissue and tumour regions_1_2, where a zero relative pressure is prescribed.

All variables at the internal boundaries between the tumour and normal tissue and between


94

different tumour regions are assumed to be continuous. The governing equations for interstitial

fluid flow are solved first, and the resolved pressure and velocity fields in tumour and normal

tissues are then used to solve the mass transfer equations for drug transport. A fixed time-step of

10 seconds is chosen following a time-step sensitivity test. The convergence criteria for residual

tolerances are 1×10-5

for momentum equations and 1×10-8

for mass transfer equations.

5.1.4. Results

A total dose of 50 mg/m2 non-encapsulated doxorubicin [3] is continuously infused within 2

hours [55], simulating a typical treatment for a 70 kg patient [51]. As a foundation for drug

delivery, the micro-mechanical environment in the tumour is examined first. This is followed by

an analysis of the concentration profiles and drug cytotoxic effect.

5.1.4.1. Interstitial Fluid Field

Interstitial fluid pressure (IFP) plays an important role in determining the transport of drug in

tumours. IFP is obtained by solving the governing equations for interstitial flow for the entire

tumour region and its surrounding normal tissue, subject to the boundary conditions described

above and a vascular pressure of 2080 Pa. As shown in Figure 5-2, there is no discernable

difference among different tumour regions, although IFP in the tumour is much higher than that

in the normal tissue. The observation that IFP is not sensitive to tumour microvasculature

distribution is consistent with the results of Baxter and Jain [50] who found that even the

presence of a necrotic core had virtually no effect on the IFP profile, unless the necrotic core size

is greater than 90% of the tumour size. This is because in regions except for a thin layer close to

the outer boundary of the tumour, IFP equilibrates with the effective vascular pressure, which is

given by ivpvp ; refer to Chapter 3 for definition of symbols and parameter values.

Based on the model parameters adopted in this study, the effective vascular pressure is 1533.88

Pa; this is reached in all tumour regions except region_1 and region_2, where the mean IFP is

slightly lower. This means that there is no net filtration between the microvessels and

interstitium in regions 3~6 which are surrounded by regions 1 and 2. On the other hand, the

rather uniform IFP in the interior of tumour implies that pressure-induced convection of

interstitial fluid is weak there, and convective drug transport in the interstitium occurs mainly

within a thin layer at the tumour/normal tissue interface, where there is a steep pressure gradient.


95

Figure 5-2. Interstitial fluid pressure distribution in tumour and normal tissues.

5.1.4.2. Doxorubicin Concentration

Figure 5-3 shows the spatial distribution of free doxorubicin at different time points in the

tumour. It is clear that the extracellular concentration of doxorubicin is non-uniform, which is

not surprising since the transport of drug is channelled through blood vessels, and as such,

vasculature density is expected to have a strong influence on the spatial distribution of drug.

Since the difference in extra- and intra-cellular concentrations between sub-regions 1_1 and 1_2

is very small, these sub-regions are combined and referred to as region_1 in the following

discussion.

1 hour 3 hour 6 hour 12 hour

Figure 5-3. Spatial distribution of free doxorubicin extracellular concentration in tumour regions (total dose = 50

mg/m2, infusion duration = 2 hours)

Figure 5-4 shows the variation of extracellular concentration of free and bound doxorubicin in

the liver tumour and each of its sub-regions of different vessel densities. In all regions, the

predicted concentration increases during the infusion period, when doxorubicin is administrated

continuously into the circulatory system. After the end of administration, both free and bound

1500.0

1236.2

1018.8

839.6

691.9

570.2

470.0

387.2

319.2

263.0

216.8

178.7

147.2

121.3

100.0

IFP (Pa)

8.25×10-5

6.63×10-5

5.33×10-5

4.28×10-5

3.44×10-5

2.76×10-5

2.22×10-5

1.79×10-5

1.44×10-5

1.15×10-5

9.27×10-6

7.45×10-6

Cfe (kg/m3)


96

doxorubicin concentrations start falling immediately. There are obvious differences in the drug

concentrations in the different tumour regions until at approximately 12 hours, when uniform

doxorubicin concentrations are reached in all tumour regions (also demonstrated in Figure 5-3).

The rate of change of doxorubicin concentration is slower in regions with low vessel density (e.g.

region_6), owing to reduced drug exchange between plasma and interstitial fluid as a result of

sparse vasculature.

0 1 2 3 4 5 6 7 8 9 10 11 12

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Region_1

Region_2

Region_3

Region_4

Region_5

Region_6

Fre

e D

oxoru

bic

in E

xtr

acel

lual

r C

once

ntr

atio

n

(kg/m

3)

Time (hr)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12

0.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

(kg/m

3)

Time (hr)

Region_1

Region_2

Region_3

Region_4

Region_5

Region_6

(b)

Figure 5-4. Spatial mean doxorubicin extracellular concentration in tumour as a function of time. (a) Free

doxorubicin, and (b) bound doxorubicin. (total dose = 50 mg/m2, infusion duration = 2 hours)

Although vascular densities vary in the different tumour regions, the results in each tumour

region show that nearly 75% doxorubicin is in bound form, and both free and bound drugs share

a similar time course, confirming that the mechanism of binding with protein is independent of

vasculature distribution.

0 1 2 3 4 5 6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Region_1

Region_2

Region_3

Region_4

Region_5

Region_6

Doxoru

bic

in I

ntr

acel

lula

r C

once

ntr

atio

n

(ng/1

05ce

lls)

Time (hr)

Figure 5-5. Spatial mean of doxorubicin intracellular concentration in tumour regions as a function of time. (total

dose = 50 mg/m2, infusion duration = 2 hours)


97

Intracellular concentration also displays a non-uniform pattern among the tumour regions of

different vessel densities. As shown in Figure 5-5, the rate of change of intracellular

concentration is slower and peak concentration is lower in poorly vascularised regions, similar to

the pattern observed for extracellular concentration. This is because cellular uptake follows the

process of transvascular transport and migration in interstitial space, therefore, the intracellular

concentration profile is strongly dependent on the concentration in extracellular space. The

highest peak is observed in the region with the densest microvasculature, and results in Figure 5-

5 indicate that the peak value increases monotonically with vascular density.

5.1.4.3. Doxorubicin Cytotoxic Effect

The predicted anticancer effectiveness, assessed based on the variation in the percentage of

tumour cells (Figure 5-6), suggests that cell killing is more effective in well-vascularised regions

in the first few hours, but the trend is reversed thereafter. However, the difference among the

various regions is small, with the maximum difference being 3% between region_1 (high vessel

density) and region_6 (low vessel density) at 12-hour following drug administration.

0 2 4 6 8 10 12

75

80

85

90

95

100

Region_1

Region_2

Region_3

Region_4

Region_5

Region_6

Surv

ival

Fra

ctio

n o

f T

um

our

Cel

ls (

%)

Time (hr)

Figure 5-6. Spatial mean percentage of tumour cells in tumour regions as a function of time (total dose = 50 mg/m2,

infusion duration = 2 hours).

The spatial distributions of survival cell fraction in the tumour regions at different time points are

shown in Figure 5-7. As a result of the non-uniform distribution of extracellular concentration

(see Figures 5-3 and 5-4), different treatment outcomes are found in different tumour regions.


98

98.7

95.4

92.0

88.6

85.2

81.8

78.4

75.0

f (%)

1 hour 3 hours 6 hours 12 hours

Figure 5-7. Spatial distribution of survival cell fraction in tumour regions.

(total dose = 50 mg/m2, infusion duration = 2 hours)

5.1.4.4. Comparison of Various Administration Schedules

Comparisons between different doses (25, 50 and 75 mg/m2) for a 2-hour continuous infusion

confirm that the qualitative trend presented above is not altered by changes in dose level. As

shown in Figure 5-8, the maximum differences in free doxorubicin extra- and intra-cellular

concentrations among tumour regions increase proportionally with an increase in drug dose.

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

75 mg/m2

50 mg/m2

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

(kg/m

3)

25 mg/m2

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Intr

acel

lula

r C

once

ntr

atio

n (

ng/1

05ce

lls)

25 mg/m2

50 mg/m2

75 mg/m2

(b)

Figure 5-8. Maximum difference among tumour regions over treatment with various doses. (a) free doxorubicin

extracellular concentration and (b) intracellular concentration (infusion duration = 2 hours)

Comparisons are also made between different infusion durations (1-hour, 2-hour and 4-hour) for

a 50 mg/m2 continuous infusion, and the qualitative tread among tumour regions is not altered by

this parameter. The maximum differences in free doxorubicin extra- and intra-cellular

concentration among tumour regions, shown in Figure 5-9, indicate that prolonging the infusion

duration may help to reduce the difference in both extra- and intra-cellular concentration among

tumour regions.


99

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

(kg/m

3)

1 hour 2 hour 4 hour

(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Intr

acel

lula

r C

once

ntr

atio

n (

ng/1

05ce

lls)

1 hour 2 hour 4 hour

(b)

Figure 5-9. Maximum difference among tumour regions over treatment with various infusion durations. (a) free

doxorubicin extracellular concentration and (b) intracellular concentration (total dose = 50 mg/m2)

5.2. Tumour Size

The effect of tumour size is investigated using three prostate tumours under free form drug

delivery. The drug transport processes and their governing equations are consistent with those

adopted in Chapter 4. The unique tumour geometry and parameters are described below.


Three prostate tumour models are reconstructed from images acquired from patients using a 3.0-

Tesla MR scanner (DISCOVERY MR750, GE, Schenectady, New York, USA). Multislice

anatomical images of the prostate were acquired in three orthogonal planes with echo-planer (EP)

sequence, with each image comprising 256 by 256 pixels. Other imaging parameters are given in

Table 5-3.


Pixel Size

(mm)

Field of View

(cm)

Slice Thickness

(mm)

Repetition Time

(ms)

Echo Time

(ms)

case_1

1.250 32.0 7.000

3675 85.7

case_2 4000 84.4

case_3 4000 91.5

An example of the MR images has been shown previously in Figure 4-1. Transverse images are

processed using image analysis software Mimics (Materialise HQ, Leuven, Belgium), and the

tumours are segmented from its surrounding normal tissues based on signal intensity values. The


100

resulting smoothed surfaces of the tumour and normal tissues are imported into ANSYS ICEM

CFD to generate computational mesh for the entire volume. The dimension of the reconstructed

models shown in Figure 5-10 is approximately 40~50 mm in length for three cases. The final

mesh consists of 1,173,908, 1,163,374 and 1,474,027 tetrahedral elements for case 1, 2 and 3,

respectively. These meshes have been tested to produce grid independent solutions.

tumour normal tissue

Figure 5-10. Model geometry: (a) case_1; (b) case_2; (c) case_3


Since the time window of simulations is smaller than the growth of tumour, all the geometric and

transport parameters are considered to be independent of time. Baseline values for all model

parameters are summarized in Tables 3-1 and 3-3. The tumour size dependent parameter is the

surface area of blood vessels per unit volume of tumour tissue (S/V), which reflects the

microvasculature density.

Figure 5-11. Estimation of blood vessel surface area to tissue volume ratio of tumours with various tumour sizes

(experimental data extracted from [116]). For the best fitted curve using equation y=axb (a = 54.68, b = -0.2021).

0 200 400 600 800 1000 1200 1400 1600

10

12

14

16

18

20

22

24

26

28

S/V

(m

m-1)

Tumour Volume (mm3)

experimental data

fitting curve


101

Pappenheimer measured S/V normal tissue and found this to be 70 cm-1

[117]; this value was

adopted by Baxter and Jain [49, 50, 88] in their series of works. Hilmas and Gillette [116]

measured this ratio in tumours with different volumes and at different growth stages by using

morphometric methods. By using non-linear least squares curve-fitting technique, the

relationship between S/V and tumour size is obtained (shown in Figure 5-11), and the S/V value

for each prostate tumour is then calculated based on its own size; all the values obtained are

summarized in Table 5-4.

Table 5-4. Tumour size and the blood vessel surface area to tissue volume ratio in each model

Tumour size (mm3) S/V (mm

-1)

case_1 45.46 25.28

case_2 2277.97 11.46

case_3 7118.84 9.10

The numerical method and boundary conditions are consistent with previous studies in this

Chapter.

5.2.3. Results

A numerical simulation has been carried out for continuous infusion of 50 mg/m2 [3] non-

encapsulated doxorubicin in 2 hours [55], which corresponds to a standard treatment for a patient

of 70 kg body weight [51]. The obtained interstitial fluid field is presented first as this provides

the microenvironment for drug delivery. This is followed by comparisons of doxorubicin

concentration in the three prostate tumours to investigate the effect of tumour size on drug

delivery. Finally, predicted treatment outcomes based on survival fractions of tumour cells

calculated from the pharmacodynamics model are compared.

5.2.3.1. Interstitial Fluid Field

Because of abnormalities in tumour, such as vasculature, cellular matrix and the components of

interstitial fluid, tumour interstitial fluid pressure (IFP) has been found to be higher than that in

normal tissues [49, 50, 88, 89]. Predicted IFP contours at a representative cross-section for each

of the three tumours are shown in Figure 5-12. IFP is uniformly high in the entire tumour except

in a thin layer close to the tumour boundary. Owing to the low pressure in normal tissue, a sharp


102

pressure gradient can be found near the boundary between tumour and the surrounding normal

tissue.

148013201160100084068052036020040

IFP (Pa)

case_1 case_2 case_3

Figure 5-12. Interstitial fluid pressure distribution in tumour and normal tissue

The predicted spatial mean IFP and spatial mean transvascular flux per tumour volume in each

tumour are shown in Figure 5-13 (a) and (b), respectively. It can be clearly observed that spatial

mean IFP varies with tumour size, with higher IFP in larger tumour. Since IFP in tumour centre

are at the same for all three models (at 1533.88 Pa), the observed variation of mean IFP with

tumour size is caused by the thin layer with a sharp IFP gradient near the tumour boundary as

shown in Figure 5-12. It has been reported in a parameter sensitivity study that IFP gradient is

steeper in larger tumour [49], which would result in a higher spatial mean IFP. This is consistent

with the finding presented here.

0

100

200

300

1300

1400

1500

1600

45.46 mm3

1478.04 Pa1471.65 Pa

7118.74 mm3

2277.97 mm3

Spat

ical

-mea

n I

nte

rsti

tial

Flu

id P

ress

ure

(P

a)

1399.99 Pa

(a)

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

7.0x10-5

8.0x10-5

1.07×10-5 s

-11.50×10

-5 s

-1

Tra

nsv

ascu

lar

Flu

x p

er T

um

our

Volu

me

(s-1)

7.11×10-5 s

-1 (b)

45.46 mm3

2277.97 mm3

7118.74 mm3

Figure 5-13. The spatial mean interstitial fluid pressure and transvacular flux per tumour volume as a function of

tumour size


103

The transvascular flux per tumour volume from blood to interstitium (Fv) is governed by

Equation (3-2), which is recalled here:

ivivvv ppV

SKF (3-2)

As shown in Figure 5-13 (b), the spatial mean transvascular flux per tumour volume from

vasculature is higher in small tumour. This is because: (1) the driving force is high in small

tumour owing to the low value of spatial mean interstitial fluid pressure and uniform

intravascular pressure; (2) higher S/V in small tumour improves this exchange. However,

referring to the spatial distribution of IFP in Figure 5-12, the transvascular flux mainly occurs in

the thin layer at tumour/normal tissue interface owing to the equilibrium between IFP and

effective vascular pressure within tumour centre. Results also show that spatial mean IFP and

transvascular flux per tumour volume are non-linearly related to tumour volume.

5.2.3.2. Doxorubicin Concentration

Intravascular concentrations of free and bound doxorubicin are shown in Figure 5-14.

Doxorubicin in blood circulation system keeps increasing in the 2-hour infusion period and

reaches the peak at the end of administration. This is followed by a rapid fall, after which there is

a gradual and slow reduction of doxorubicin concentration as time proceeds. Quantitative

comparison shows that 75% doxorubicin is bound with proteins in blood stream.

0 1 2 3 4 5 6 7 8 9

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

Doxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n (

kg/m

3)

Time (hr)

free doxorubicin

bound doxorubicin

Figure 5-14. Free and bound doxorubicin concentration in blood after administration as a function of time. (infusion

duration=2 hours, total dose=50mg/m2)


104

Free and bound doxorubicin extracellular concentration in tumour and its holding normal tissue

are shown in Figure 5-15. Regardless of the tumour size, both free and bound doxorubicin

concentrations in tumour and normal tissue increase rapidly during the initial period after

administration, reach their peaks and then fall to a low level after the end of infusion. The rate of

change in doxorubicin concentration slows down with the increase in tumour size. This is

because S/V is lower in larger tumour, leading to less drug exchange between blood and tumour

interstitium. As shown in Figure 5-15 (c) and (d), variation of tumour size has no obvious

influence on drug concentration in normal tissue.

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

case_1

case_2

case_3

(a)

0 1 2 3 4 5 6 7 8 9

0.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

case_1

case_2

case_3

(b)

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

case_1

case_2

case_3

(c)

0 1 2 3 4 5 6 7 8 9

0.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

Bound D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

case_1

case_2

case_3

(d)

Figure 5-15. Spatial mean doxorubicin extracellular concentration as a function of time under 2-hour continuous

infusion, dose=50mg/m2. (a) Free and (b) bound doxorubicin in tumour, (c) bound and (d) bound doxorubicin in

normal tissue

Figure 5-16 shows the amount of drug exchange between vasculature and interstitial fluid by

convection in each tumour. Compared with the blood concentration in Figure 5-14, doxorubicin

exchange by convection follows the same pattern. This means that blood concentration has a


105

direct influence on transvascular transport of doxorubicin by convection. Result in Figure 5-16

also shows that transvascular transport in small tumour is higher than in large tumour.

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-9

4.0x10-9

6.0x10-9

8.0x10-9

Fre

e D

oxoru

bic

in E

xch

ange

bet

wee

n B

lood

and I

nte

rsti

tial

Flu

id (

kg/m

3)

Time (hr)

case_1

case_2

case_3

(a)

0 1 2 3 4 5 6 7 8 9

0.0

2.0x10-9

4.0x10-9

6.0x10-9

8.0x10-9

1.0x10-8

1.2x10-8

1.4x10-8

1.6x10-8

Bound D

oxoru

bic

in E

xch

ange

bet

wee

n B

lood

and I

nte

rsti

tial

Flu

id (

kg/m

3)

Time (hr)

case_1

case_2

case_3

(b)

Figure 5-16. Doxorubicin exchange between blood vasculature and interstitial fluid by convection as a function of

time. (a) free and (b) doxorubicin exchange (2-hour infusion, total dose =50 mg/m2)

Regardless of the tumour size, transvascular transport of doxorubicin by diffusion shown in

Figure 5-17 experiences the following stages: (1) increasing rapidly until reaching the peak, and

then decreasing slowly to a positive level at the end of administration; (2) dropping sharply to a

negative level in a few minutes and then returning very slowly to zero.

0 1 2 3 4 5 6 7 8 9

-3.0x10-6

-2.0x10-6

-1.0x10-6

0.0

1.0x10-6

2.0x10-6

3.0x10-6

4.0x10-6

5.0x10-6

case_1

case_2

case_3

Fre

e D

oxoru

bic

in E

xch

ange

bet

wee

n B

lood

and I

nte

rsti

tial

Flu

id (

kg/m

3)

Time (hr)

(a)

0 1 2 3 4 5 6 7 8 9

-3.0x10-8

-2.0x10-8

-1.0x10-8

0.0

1.0x10-8

2.0x10-8

3.0x10-8

4.0x10-8

Bound D

oxoru

bic

in E

xch

ange

bet

wee

n B

lood

and I

nte

rsti

tial

Flu

id (

kg/m

3)

Time (hr)

case_1

case_2

case_3

(b)

Figure 5-17. Doxorubicin exchange between blood vasculature and interstitial fluid as a function of time in each

tumour. (a) free and (b) bound doxorubicin exchange by diffusion owing to concentration gradient (2-hour infusion,

total dose = 50 mg/m2)

Because the initial value of doxorubicin extracellular concentration is zero, the rapid increase in

intravascular concentration results in a large concentration gradient between blood and the


106

interstitial fluid after administration begins, therefore, transvascular transport by diffusion

increases rapidly to its peak. Since the increase of blood concentration slows down, the gradient

becomes small and leads to a reduction in diffusive transport. However, it is still positive until

the end of drug administration, after which blood concentration drops rapidly and no more

doxorubicin is supplied. As a consequence, the concentration gradient reduces to zero quickly.

Since doxorubicin is washed out quickly, its extracellular concentration becomes higher than

blood concentration, causing the exchange to reverse owing to a negative concentration gradient.

Afterwards, the negative gradient becomes smaller as the extracellular concentration decreases,

causing the diffusive transport to return to zero gradually.

As shown in Figures 5-14 and 5-15, most of doxorubicin is in bound form in blood plasma and

interstitial fluid. However, comparison of trans-vascular exchange of free and bound doxorubicin

presented in Figures 5-16 and 5-17 indicates that transvascular exchange mainly depends on free

doxorubicin. This can be attributed to the small osmotic reflection coefficient and high

permeability of free doxorubicin, which ease the transvascular process. Results also show that

the rate of change slows down as the tumour size increases. This is because the sparse

microvasculature in large tumour reduces drug exchange between blood and interstitial fluid in

tumour.

Comparisons are made between spatial mean convection exchange and diffusion exchange by

applying the same administration method to case_2. The result in Figure 5-18 shows that

diffusion exchange is up to 1000 fold higher than convection during the infusion period. This

means that diffusion dominants the drug transport process in the initial hours of anticancer

treatment. After the end of administration, diffusive transport driven by concentration difference

weakens but is still dominant, resulting in continued reduction in extracellular concentration. It is

worth noting that transvascular transport by convection mainly occurs in a thin layer at the

interface between tumour and normal tissue. On the contrary, transvascular transport by diffusion

takes place in the entire tumour and normal tissue owing to the concentration gradient across

microvasculature walls.


107

0 1 2 3 4 5 6 7 8 9

-2.0x10-6

-1.0x10-6

0.0

1.0x10-6

2.0x10-6

3.0x10-6

Fre

e D

oxoru

bic

in E

xch

ange

bet

wee

n B

lood

and I

nte

rsti

tial

Flu

id (

kg/m

3)

Time (hr)

by convection

by diffusion

(a)

0 1 2 3 4 5 6 7 8 9

-1.5x10-8

-1.0x10-8

-5.0x10-9

0.0

5.0x10-9

1.0x10-8

1.5x10-8

2.0x10-8

by convection

by diffusion

Bound D

oxoru

bic

in E

xch

ange

bet

wee

n B

lood

and I

nte

rsti

tial

Flu

id (

kg/m

3)

Time (hr)

(b)

Figure 5-18. Comparison of (a) free and (b) bound doxorubicin exchange between blood and interstitial fluid by

convection and diffusion in case_2 (2-hour infusion, total dose = 50mg/m2)

Figure 5-19 presents the intracellular doxorubicin concentration in the tree tumours for 2-hour

direct infusion. Given that intracellular concentration depends on the free doxorubicin

extracellular concentration in tumour, intracellular concentration increases rapidly to its peak and

before falling off. The computational model predicts that a small tumour with denser

microvasculature has more rapid changing rates and higher peaks. However, intracellular

concentration in a large tumour sustains at a slightly higher level over time.

0 1 2 3 4 5 6 7 8 9

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Doxoru

bic

in I

ntr

acel

lula

r C

once

ntr

atio

n i

n T

um

our

(ng/1

05ce

lls)

Time (hr)

case_1

case_2

case_3

Figure 5-19. Temporal profiles of predicated intracellular concentration in tumours with different sizes

(2-hour infusion, total dose = 50 mg/m2)


Figure 5-20 presents the fraction of survival tumour cells by applying the pharmacodynamics

model described by Equation (3-30). In small tumours with a denser vasculature, more tumour


108

cells can be killed in the first few hours after administration begins. However, the drug cytotoxic

effect in small tumours becomes worse as time proceeds. This is because the intracellular

concentration in small tumours increases rapidly to a higher peak value, but also decreases faster

to a low level (shown in Figure 5-19). This fast reduction leads to insufficient cell killing in the

later hours of a treatment in small tumours. As can be observed, differences in cytotoxic

effectiveness are limited, with a difference of up to 3.2% between case_1 (small tumour) and

case_3 (large tumour) in the simulation duration.

0 1 2 3 4 5 6 7 8 9

75

80

85

90

95

100

Surv

ival

Fra

ctio

n o

f T

um

our

Cel

ls (

%)

Time (hr)

case_1

case_2

case_3

Figure 5-20. Survival cell fraction of tumour as a function of time in three tumours with different sizes (2-hour

infusion, total dose = 50 mg/m2)

5.3. Discussion

Results from the mathematical model show that IFP is uniform within a tumour except for the

tumour boundary because IFP in the tumour interior reaches equilibrium with the effective

vascular pressure, implying that there is no net filtration between the blood vessels and tumour

interstitium. Therefore, convective transvacular transport of drug only occurs within a thin layer

at the tumour/normal tissue interface, where there is a steep pressure gradient, and hence

transvascular transport mainly replies on diffusion driven by concentration gradient across

microvasculature wall.

After administration, doxorubicin accumulates faster in tumour regions with denser vasculature

to achieve a higher peak. This is because the rate of transvascular transport is higher than that in

tumour regions with sparser microvasculature. Similarly, doxorubicin is rapidly drained back to

blood lumen after the end of administrations in these regions owing to the fast plasma clearance


109

and reverse concentration gradient, resulting in a quick reduction of concentration. Similar

results have been found when examining the effect of tumour size with different

microvasculature density. The rate of change of concentration is faster in small tumours with

dense microvasculature, where the anticancer efficacy is reduced.

Results demonstrate that vasculature in tumour is a crucial factor in drug delivery. Nevertheless,

the mathematical model employed here involves a number of assumptions. First, the tumour

vascular network is treated as a distributed source term in the governing equations instead of

being modelled explicitly. As a result, the effects of intravascular transport and geometric

features of tumour vasculature are not included. While this may be an over-simplification, it is

practical given the lack of information on realistic tumour vasculature geometry and the

complexity in direct modelling of blood flow in a vascular network coupled with interstitial drug

transport. Second, all the tumour regions are assumed to be perfused via microvasculature in the

studied liver tumour model. Drug transport in tumour regions without functional blood vessels,

such as necrotic cores, is not investigated. Third, a 2D tumour model is used instead of a full 3D

model. Based on the comparisons between 2D and 3D models of a prostate tumour discussed in

Chapter 4, the 3D effect on spatial-mean parameters, such as intracellular drug concentration and

tumour cell density, is negligible. This is consistent with the work of Teo et al [79] who found

qualitatively similar results between their 2D and 3D models of a brain tumour.

Other assumptions include: (1) signal intensities of tracer Gd-DTPA in post-contrast MR images

are used to calculate vascular density, with the tracer concentration in the plasma being assumed

to be proportional to the relative change in signal intensity. This could be improved by

measuring the intravascular concentration of Gd-DTPA directly in MR imaging [136]. (2)

uniform transport properties and uniform tumour cell density are assumed in each tumour region.

(3) the intratumoural formation of new blood vessels is a dynamic and complex process

involving several cellular and subcellular events in vivo [137], and the administrated doxorubicin

can inhibit angiogenesis rather than cytotoxic effect on tumour cells in doxorubicin-insensitive

tumour models [138-140]. Changes in blood vessels are expected to directly affect drug

concentration in interstitial fluid and cell killing. However, since these variations usually take

place in days [137, 138] which are much longer than the simulated time window, the


110

vasculature-related parameters are assumed to be constant in this series of work, but their time-

varying nature needs to be considered when long simulations are carried out.

5.4. Summary

The effects of vasculature-related microenvironment in tumour have been investigated.

Modelling of a liver tumour with heterogeneous vasculature distribution shows that the tumour

interstitial fluid pressure is insensitive to the distribution of blood vessels. Even in parts of the

tumour that is not well surrounded by holding normal tissue, the pressure gradient there is still

limited to a thin layer with a negligible impact on the tumour interior. Better anticancer

effectiveness is achieved in regions with low vasculature density, owing to the reduced loss of

drugs to the plasma.

Modelling of three prostate tumours with different sizes under direct continuous infusion

demonstrates the nonlinear relationships between spatial-mean interstitial fluid pressure and

tumour volume, as well as between transvascular flux per tumour volume and tumour volume.

During the infusion period, transvascular drug exchange mainly depends on diffusion, owing to

the concentration gradient of free drug between blood and interstitial fluid.

6. Thermo-Sensitive Liposome-Mediated

Drug Delivery to Solid Tumour

Thermo-sensitive liposome-mediated drug delivery is a promising method to reduce

the risk of side effects in anticancer treatment. For well-designed thermo-sensitive

liposomes (TSL) that are stable in blood, encapsulated drugs cannot be released until

local temperature is raised above the phase transition temperature of the liposome

membrane. Among several heating methods that are currently available, high

intensity focused ultrasound (HIFU) is favoured for its non-invasive nature and

accuracy when used under the guidance of MRI. In the focus region of HIFU, both

tumour and blood are heated directly by the absorbed acoustic power, whereas

temperature elevation in the surrounding region is achieved via heat transfer through

convection and conduction.

In this chapter, the acoustic model for HIFU described in Chapter 3 is firstly

validated by comparison with experimental data. Subsequently, the coupled bioheat

transfer and drug transport models are used to investigate TSL-mediated drug

delivery to a realistic 2-D prostate tumour. Finally, a comparison of temperature

profiles between 2-D and 3-D tumour models is presented.

6. THERMO-SENSITIVE LIPOSOME-MEDIATED DRGU DELIVERY TO SOLID TUMOUR

112

6.1. Acoustic Model for High Intensity Focused Ultrasound

Given that temperature elevation and the following drug release are consequences of absorption

of ultrasound energy in tissue and blood, the acoustic pressure generated by HIFU is an essential

input to the simulation of TSL-mediated drug delivery. The acoustic HIFU model described in

Chapter 3 is applied to a standard test case for which experimental measurements are available.

1.17

1.01

0.85

0.69

0.54

0.39

0.24

0.08

Acoustic Pressure

(MPa)

Figure 6-1. Spatial distribution of acoustic pressure

The test case is for a single element HIFU transducer with a 120-mm focal length and 120-mm

aperture. The transducer with a fixed frequency of 1.33 MHz is driven by a continuous wave to

generate 1.45 MPa at the focus point, with a corresponding intensity of 141.78 W/cm2. Moving

the transducer along the beam and its orthogonal directions, the acoustic pressure in water was

measured by a chosen 0.4 mm hydrophone in Solovchuk’s experiment [114]. The validation

study of acoustic pressure is based on a square geometry (8cm×5.5cm) with the focus region

located in the centre. The spatial distribution of acoustic pressure is shown in Figure 6-1.

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Numerical Result

Experimental Result

Norm

aliz

ed A

coust

ic P

ress

ure

Axial Distance (mm)

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Numerial Result

Experimental Result

Norm

aliz

ed A

coust

ic P

ress

ure

Radial Distance (mm)

(b)

Figure 6-2. Measured and computed acoustic pressure profiles at 1.33 MHz and 1.45 MPa pressure at the focal point.

(a) along axis and (b) radial distance. Experimental data are extracted from [114].


113

Normalized by the pressure at the focal point, acoustic pressure along the axial and radial

distance in the focus plane is shown in Figure 6-2. Results show that the predicted acoustic

profiles agree well with the experimental data, demonstrating the validity of the HIFU model for

prediction of acoustic profiles in biological media [114].

6.2. Thermo-Sensitive Liposome-Mediated Delivery

The combined HIFU, bioheat transfer and drug transport models are applied to a prostate tumour,

which is heated simultaneously with the administration of TSL encapsulated doxorubicin. By

solving the governing equations described in Chapter 3, spatial and temporal distributions of

temperature and drug concentration in blood, tumour and normal tissues are obtained. The

expected therapeutic efficacy is evaluated by tumour cell survival fraction.

6.2.1. Model Description

The mathematical model for TSL-mediated drug delivery consists of two parts: bioheat transfer

model and drug transport model. The former covers acoustic energy generated by a single

element ultrasound transducer, heat absorption by blood, tumour and normal tissues, as well as

heat transfer between these compartments; while the later describes the transport of encapsulated,

free and bound doxorubicin in blood, extracellular space of tumour and surrounding normal

tissues, as well as cellular uptake of the drug.

6.2.1.1. Model Geometry

A 2-D model of prostate tumour is reconstructed from MR images; the same model has been

used in Chapter 4. As shown in Figure 6-3, two focus regions are chosen in order to achieve

adequate coverage of the tumour [76, 77].

Figure 6-3. Model geometry and locations of focus regions.


114

ANSYS ICEM CFD is used to generate the computational mesh. The final mesh consists of

64,966 triangular elements. This is obtained based on mesh independence tests which show that

the difference in predicted drug concentration between the adopted mesh and a 10 times finer

mesh is less than 1%.

6.2.1.2. Model Parameters

Parameters describing the basic physical and transport properties of doxorubicin, tumour and

normal tissue have been given in Tables 3-1, 3-2 and 3-3 in the preceding chapter. Additional

parameters required for the acoustic and bioheat transfer models are summarised in Table 6-1.

Since temperature elevations in response to HIFU heating may influence some of the properties

used in the drug transport model, temperature dependence of these properties is also considered.

Table 6-1. Acoustic and thermal properties

Symbol Description Unit Value

Source Blood Tumour Normal Tissue

va ultrasound speed m/s 1540 1550 1550 [74]

ρ density kg/m3 1060 1000 1055 [74]

c specific heat J/kg/K 3770 3800 3600 [74]

k thermal conductivity W/m/K 0.53 0.552 0.512 [74]

α absorption coefficient 1.5 f 9 f 9 f [74]

wb0 perfusion rate of blood flow at 37oC s-1 - 0.002 0.018 [45]

R universal gas constant J/mol/K - 8.314 8.314 [141]

ΔE activation energy J/mol - 6.67×105 6.67×105 [141]

Af frequency factor s-1 - 1.98×10106 1.98×10106 [141]

* f represents ultrasound frequency

HIFU Transducer

The acoustic source is a single-element, spherically focused transducer, and its parameters are

summarized in Table 6-2. The local acoustic intensity and peak negative pressure are set as 120

W/cm2 and 1.98 MPa, respectively.

Table 6-2. HIFU transducer parameters

Parameter

(unit)

Inside diameter

(mm)

Outside diameter

(mm)

Focal length

(mm)

Frequency

(MHz)

20.0 70.0 62.64 1.10


115

Temperature controller

In TSL-mediated drug delivery, it is desirable to keep the tumour temperature within a narrow

hyperthermia range which should not exceed 43oC to avoid vascular shut down [142]. The phase

transition temperature of liposome strongly depends on its formulation, and a typical temperature

is 42oC based on experimental studies [69, 143]. In order to achieve the target temperature range,

a feedback system with an appropriate temperature control mechanism is required. In the present

study, a temperature controller is incorporated, which is designed to adjust the applied ultrasound

power according to the maximum temperature in the heated region (Tmax). The transducer power

is determined by

tTTktTTktP ettciettcp maxargmaxarg (6-1)

where kcp and kci are Proportional-Integral (PI) control parameters, which are set as 5 and 0.006

respectively, based on values in the literature [72] and previous studies.

Permeability

(1) Free & bound doxorubicin

Dalmark and Strom [144] measured the permeability of free doxorubicin at various temperatures,

and found that the logarithmic value of permeability is proportional to temperature. Based on

their data, a linear relationship was found between fold increase in logarithmic value of

permeability and temperature, as given by Equation (6-2), and the best fitting line is shown in

Figure 6-4.

21log zTzP (6-2)

where z1 and z2 are fitted parameters which are 0.07 and 2.43, respectively.

Hence the fold increase in permeability is

211102

1 TTz

PP

(6-3)

The permeability of bound doxorubicin is assumed to follow the same relation.


116

Figure 6-4. Permeability of free doxorubicin as a function of temperature. Curve fitting parameters z1=0.07 and z2=-2.43.

Experimental data extracted from [144].

(2) Liposome Encapsulated Doxorubicin

For a baseline temperature of 34oC, extracellular concentrations of TSL doxorubicin were found

to have increased by 76-fold and 38-fold upon heating to 45˚C and 42˚C, respectively [145].

Based on these experimental data [145], a relationship between fold increase in permeability and

temperature can be obtained by curve fitting, and the best fitting curve is given by Equation (6-4)

and shown in Figure 6-5.

210120 pp mTT21p

mTT11p0pp ememmm

(6-4)

where T and T0 represent temporal temperature and body temperature, respectively, and the

other terms in Equation (6-4) are curve fitting parameters.

Figure 6-5. Fold increase in permeability of liposome-mediated doxorubicin as a function of temperature. Curve

fitting parameters: mp0=-10.54, To=34, mp11=5.76, mp21=5.78, mp12=5.44, and mp22=5.46. Experimental data of [145]

are adopted.

36 37 38 39 40 41 42 43 44 45 46

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Logar

ithm

ic V

alue

of

Per

mea

bil

ty (

×10

-7m

/s)

Temperature (oC)

Experimental Data

Fitting Curve

33 34 35 36 37 38 39 40 41 42 43 44 45 46

0

10

20

30

40

50

60

70

80

90

Fold

Incr

ease

in P

erm

eabil

ity

Temperature (oC)

Experimental Data

Fitting Curve


117

Diffusivity

According to the Stokes-Einstein equation (Equation 3-48), the fold increase in diffusivity as a

function of temperature is

1

*

2

2

*

1

2

1

T

TD

D (6-5)

where D and strands for diffusivity, and μ is the viscosity of solvent. Subscripts 1 and 2

correspond to conditions 1 and 2, respectively. Owing to the lack of relevant data, viscosity

values in Equation (6-5) are assumed to be those of water [146], whose dependence on

temperature is given by Equation (6-6), with the best fitting result shown in Figure 6-6.

2

321exp TmTmmm dddd (6-6)

where T is temporal temperature, and the other terms are parameters used in curve fitting.

Figure 6-6. Viscosity of water as a function of temperature. Curve fitting parameter md1=5.10 md2=-0.03

md3=1.04×10-4

Experimental data extracted from [146]

Rate of Transmembrane Transport

The transmembrane parameter was determined by El-Kareh and Secomb [55] by curve fitting to

data obtained from in vitro experiments [101]. It has been suggested that increased cellular

uptake of doxorubicin with heating will lead to improved outcome when the drug is

administrated simultaneously with hyperthermia [55]. Based on data in [147], there is a 2.2-fold

increase at 42˚C. Here, the fold increase at temperature T is obtained by linear interpolation.

28 32 36 40 44 48 52 56 60 64

40

45

50

55

60

65

70

75

80

85

Wat

er V

isco

city

(10

-5P

as)

Temperature (oC)

Experimental Data

Fitting Curve


118

88.724.0 Tktv (6-7)

Perfusion rate

Blood perfusion rate (w) also depends on temperature [141] by the following relation:

DSww 0 (6-8)

where w0 represents the time dependent perfusion at 37˚C, and DS is the degree of vascular stasis

with a value between 0 and 1.

ttRT

E

f deADS0

*exp (6-9)

where T* is the absolute temperature as a function of time t, R is the universal gas constant, and

Af and ΔE represent the frequency factor and the activation energy, respectively.

Plasma Pharmacokinetics of Encapsulated Doxorubicin

After intravenous administration, liposome-encapsulated doxorubicin begins to circulate to all

over the body in circulatory system. Because of the continuous clearance in blood stream and

drug release triggered by environmental temperature, the intravascular concentration of

encapsulated doxorubicin (Clp) is governed by [13]:

lprellplp

lpCkCCL

dt

dC (6-10)

where CLlp and krel refer to the clearance rate and drug release rate, respectively. The initial

concentration of encapsulated doxorubicin is 0.0191 kg/m3, corresponding to a total dose of 50

mg/m2 in literature [62].

Drug Release Rate from Liposome Encapsulation

TSL is designed to release its contents rapidly upon heating [65]. The exact release rate varies

according to the composition of liposome, its preparation procedure and heating temperature [69].

The relation between percentage release and exposure time is found to follow the first-order

kinetics expressed as [148]:

tk

creleRtR

1% (6-11)


119

where %R(t) is the percentage of drug released at exposure time t, Rc is the total percentage of

drug released at a given heating temperature. This equation is used to fit experimental data

obtained in the range of 37~42˚C [143], and curve fitting for drug release at 37 o

C and 42oC is

shown in Figure 6-7.

Figure 6-7. Drug release at (a) 37oC and (b) 42

oC. Experimental data extracted from [143].

Note that the TSL formulation used in [143] still allows some drug to be slowly released at 37 oC,

although release at 42oC is much faster. From the best fitting results, release rates at different

temperatures in the range of 37~42˚C are summarized in Table 6-3. Linear interpolation is

performed to obtain release rate at temperatures between the listed temperature points, and the

release rate is assumed to be constant when temperature exceeds 42˚C.

Table 6-3. Release rates at various temperatures

T (oC) 37 38 39 40 41 42

krel (s-1

) 0.00417 0.00545 0.01492 0.02815 0.04250 0.05409

6.2.1.3. Numerical Methods

The mathematical models are implemented into a finite-volume based computational fluid

dynamics (CFD) code, ANSYS FLUENT. The User Defined Scalar (UDS) function is used to

code bioheat transfer model, mass transfer equations governing the drug transport and

pharmacodynamics. The momentum, drug transport and heat transfer equations are discretised

by using the 2nd order UPWIND scheme, and the SIMPLEC algorithm is employed for pressure-

velocity coupling. The convergence is controlled by setting residual tolerances of the momentum

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

5

10

15

20

25

30

35

40

Rel

ease

Per

centa

ge

(%)

Time (s)

Experimental Data

Fitting Curve

(a)

0 50 100 150 200 250 300 350

0

20

40

60

80

100

120

Rel

ease

Per

centa

ge

(%)

Time (s)

Experimental Data

Fitting Curve

(b)


120

equation, drug transport equations and bioheat transfer equations to be 1×10-5, 1×10-8 and 1×10-

10, respectively. As shown in Figure 6-8, interstitial fluid equations are solved first to obtain a

steady-state solution in the entire computational domain. The obtained pressure and velocity are

then introduced at time zero for the simulation of bioheat transfer, drug transport and cellular

uptake. The 2nd order implicit backward Euler scheme is employed to perform the temporal

discretization. A fixed time step size of 0.1 second is chosen after time-step sensitivity tests.

Solving continuity and momentum

equations to obtain steady-state

solution of interstitial fluid field

IFP

IFV

Solving mass

transfer equation

Solving acoustic equation to

generate the acoustic pressure

field in the whole domain

acoustic pressure

Solving bio-heat

transfer model

coupled

Temperature profiles

Drug concentration

Tumour Cell Density

Figure 6-8. Numerical Procedure

Given the time scale of modelling is much shorter than that of tumour growth, all the boundaries

of the tumour and normal tissue are assumed to be fixed. The interface between tumour and

normal tissue is treated as an internal boundary, where all variables are continuous. The relative

pressure and temperature at the surface of normal tissue is specified to be 0 Pa and 37oC, with

zero flux of drug.

6.2.2. Results and Discussion

A coupled mathematical model is used to examine the interplays among the various steps

involved in the delivery of TSL-mediated chemotherapy. Basic features of the interstitial fluid

field are the same as those discussed in Chapters 4 and 5, which will not be repeated here.

Results presented below are focused on spatial and temporal distributions of temperature and

drug concentration in blood, tumour and normal tissue.


121

6.2.2.1. Temperature Profile

For clinical application, HIFU heating for 30~60 minutes after administration of TSL are

recommended [142]. In an experimental study [149], 10 minutes HIFU heating which was

interleaved with a 5 minutes cooling down period was repeated 3 times. Therefore, simulations

have been carried out for TSL-mediated drug delivery with simultaneously HIFU heating for

durations of 10 minutes, 30 minutes and 60 minutes, respectively. Results are compared with a

control case without heating. The two focused regions presented in Figure 6-3 are heated

sequentially during the treatment, and the sonication duration for each region is 1 second, which

is chosen based on previous studies.

The time course of temperature with 10 minutes heating is shown in Figure 6-9. Variations of

maximum and spatial-mean temperature in blood, tumour and normal tissues have a similar

pattern in response to the heating schedule. The maximum temperature in tumour reaches the

target value of 42oC rapidly, and is then maintained at this level owing to the temperature control

mode governed by Equation (6-1) until the heating period ends. This is followed by a fall in

temperature during the cooling down stage. The lower temperature in blood is because the

continuous perfusion of blood is effective in removing heat from the focus region, and this

forced convection via blood is the main mechanism of heat transfer in tumour and normal tissues.

Results show that temperatures fall back to body temperature within 20 minutes after the

cessation of heating.

0 5 10 15 20 25 30

37

38

39

40

41

42

43

Max

imum

Tem

per

ature

(oC

)

Time (min)

Tumour

Normal Tissue

Blood in the Entire Domain

(a)

0 5 10 15 20 25 30

37.0

37.5

38.0

38.5

39.0

39.5

40.0

40.5

Spat

ial

Mea

n T

emper

ature

(oC

)

Time (min)

Tumour

Nomral Tissue

Blood in Tumour

Blood in Normal Tissue

(b)

Figure 6-9. Variations of temperature in tumour, blood and normal tissue as a function of time for 10 minutes

heating. (a) Maximum and (b) spatial-mean temperature.


122

Since the focus regions for direct heating are located inside the tumour (shown in Figure 6-3),

temperature in tumour increases faster and is higher than in normal tissues, whereas, temperature

elevation in normal tissues depends on heat transfer from tumour, which in turn causes heat loss

hence reduction in tumour temperature. However, since heat is continuously carried away by the

circulating blood, blood temperature in the tumour region is always lower than tumour

temperature. The large difference in spatial-mean temperature between tumour and normal

tissues (shown in Figure 6-9 (b)) indicates that the release of doxorubicin mainly occurs in the

tumour region, thereby achieving localised treatment and reducing the risk of side effects. Since

blood in the tumour region is directly heated by HIFU, its mean temperature is higher than that in

the surrounding normal tissue.

Spatial distributions of temperature in tissues and blood at 10 minutes after heating are shown in

Figure 6-10. It is clear that the highest temperature is located in the focus regions where tumour

and blood are directly heated by HIFU. Temperature elevation in the surrounding regions is a

consequence of convective and conductive heat transfer from the focus regions, and temperature

in normal tissues is mostly around 37oC, lower than in the tumour. This temperature distribution

can successfully enhance doxorubicin release in tumour.

41.6

41.0

40.3

39.7

39.0

38.3

37.6

37.0

T (oC)

38.6

38.3

38.1

37.9

37.7

37.5

37.2

37.0

T (oC)

Tissue Temperature Blood Temperature

Figure 6-10. Spatial distribution of temperature in (a) tissue and (b) blood at 10min after heating starts.

Figure 6-10 (a) also shows that tissue temperature varies gradually from tumour to normal tissue,

owing to similar thermal properties of tumour and normal tissues summarized in Table 6-1. The

temperature distribution in blood shown in Figure 6-10 (b) has a similar pattern to tissue

temperature but with a narrower range, so that the intravascular release of doxorubicin is much

higher in HIFU focus regions than in the periphery.

Temporal variations of temperature for 30 and 60 minutes heating are shown in Figure 6-11.


123

Identical shapes can be found in each compartment for a given heating duration. The spatial

mean temperature of tumour tissue remains at 40oC, which is lower than the most effective

release temperature.

0 10 20 30 40 50 60

37

38

39

40

41

42

43

Max

imum

Tem

per

ature

(oC

)

Time (min)

Tumour

Normal Tissue


(a)

0 10 20 30 40 50 60

37.0

37.5

38.0

38.5

39.0

39.5

40.0

40.5

Spat

ial

Mea

n T

emper

ature

(oC

)

Time (min)

Tumour

Normal Tissue

Blood in Tumour


(b)

0 10 20 30 40 50 60 70 80 90

37

38

39

40

41

42

43

44

Tumour

Normal Tissue


Max

imum

Tem

per

ature

(oC

)

Time (min)

(c)

0 10 20 30 40 50 60 70 80 90

37.0

37.5

38.0

38.5

39.0

39.5

40.0

40.5

41.0S

pat

ial

Mea

n T

emper

ature

(oC

)

Time (min)

Tumour

Normal Tissue

Blood in Tumour


(d)

Figure 6-11. Variations of temperature in tumour, blood and normal tissue as a function of time for various heating

durations. (a) Maximum and (b) spatial-mean temperature for 30 minutes heating; (c) Maximum and (d) spatial-

mean temperature for 60 minutes heating.

6.2.2.2. Drug Concentration

Doxorubicin is initially encapsulated in TSL which are administrated intravenously and carried

by the blood stream into the tumour and its surrounding normal tissue. The release of

doxorubicin from TSL is triggered by elevated temperature upon HIFU heating. Therefore, the

concentration of encapsulated and released doxorubicin in blood should be examined when

evaluating its anticancer efficacy. The time courses of encapsulated doxorubicin intravascular

concentration with various heating durations are compared in Figure 6-12. Because of the short

administration period which can be ignored in comparison with the heating duration, drug


124

concentration achieves its peak at the very beginning of treatment. Owing to rapid clearance in

the circulatory system and increased release as temperature rises, the intravascular concentration

of encapsulated doxorubicin falls continuously until it drops to 0.

0 5 10 15 20 25 30

0.0

5.0x10-3

1.0x10-2

1.5x10-2

2.0x10-2

Enca

psu

late

d D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (min)

No-Heating

10 min Heating

30 min Heating

60 min Heating

9 10 11 12 13 14 15 16 17 18

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

Enca

psu

late

d D

oxoru

bic

in I

ntr

avas

cula

r

Conce

ntr

atio

n i

n T

um

our

(kg/m

3)

Time (min)

(a)

0 5 10 15 20 25 30

0.0

5.0x10-3

1.0x10-2

1.5x10-2

2.0x10-2

No-Heating

10 min Heating

30 min Heating

60 min Heating

Enca

psu

late

d D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (min)

9 10 11 12 13 14 15 16 17 18

0.0

3.0x10-4

6.0x10-4

9.0x10-4

1.2x10-3

1.5x10-3

1.8x10-3

Enca

psu

late

d D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (min)

(b)

Figure 6-12. Spatial-mean concentration of encapsulated doxorubicin in blood stream as a function of time (dose =

50 mg/m2). (a) in tumour, and (b) in normal tissue.

Compared with the control case without HIFU heating, encapsulated doxorubicin concentration

in tumour (Figure 6-12 (a)) reduces faster upon HIFU heating, owing to enhanced release rate in

response to temperature raise. It appears that encapsulated doxorubicin concentration drops to

almost zero after about 20 minutes, and the duration of heating (10, 30 and 60 mins) has little

effect on encapsulated doxorubicin concentration in tumour. Since normal tissue temperature is

approximately 2~3 degrees lower than tumour temperature during HIFU heating, the effect of

heating on encapsulated doxorubicin concentration in normal tissue is trivial (Figure 6-12 (b)).

1.23×10-3

1.00×10-3

0.78×10-3

0.55×10-3

0.33×10-3

0.10×10-3

C (kg/m3)

Figure 6-13. Encapsulated doxorubicin concentration in blood at 10 min after heating

The spatial distribution of encapsulated doxorubicin concentration in blood is shown in Figure

6-13 at 10 minutes after heating; it clearly demonstrates that there is little encapsulated

doxorubicin left in and around HIFU focus regions.


125

The time courses of encapsulated doxorubicin concentration in tumour extracellular space are

presented in Figure 6-14 (a) for various heating durations. Compared with the case without

heating, encapsulated doxorubicin accumulates faster in the extracellular space of the tumour as

soon as HIFU heating begins. This is attributed to enhanced vascular permeability in response to

temperature elevation. However, the peak concentration is lower and occurs earlier in treatments

with hyperthermia. The effect of the heating duration can only be noticed after heating ends.

0 5 10 15 20 25 30

0.0

2.0x10-5

4.0x10-5

6.0x10-5

8.0x10-5

1.0x10-4

1.2x10-4

No-Heating

10 min Heating

30 min Heating

60 min Heating

Enca

psu

late

d D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (min)

(a)

0 5 10 15 20 25 30

-5.0x10-4

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

Enca

psu

late

d D

oxoru

bic

in F

lux (

kg/m

3s)

Time (min)

Net Gain

Source: Transvascular Exchange

Sink: Drug Release

(b)

27.0 27.5 28.0 28.5 29.0 29.5 30.0-2.0x10

-7

0.0

2.0x10-7

4.0x10-7

Enca

psu

late

d D

oxoru

bic

in F

lux (

kg/m

3s)

Time (hr)

Figure 6-14. Spatial-mean concentration (a) and flux (b) of encapsulated doxorubicin in tumour extracellular space

as a function of time.

Encapsulated doxorubicin concentration in tumour extracellular space depends on the balance

between the source term which accounts for drug transport from blood vessels, and the sink term

which is related to drug release from TSL. The time courses of each term and net grain in the 60-

minute heating period are presented in Figure 6-14 (b). The source term reaches its peak very

rapidly and then falls off as the intravascular concentration reduces. As a result of increased

extracellular concentration and enhanced release rate, the sink term increases initially until it

catches up with the source term at around 2 minutes after treatment begins, leading to zero

followed by negative net gain. As a consequence, encapsulated doxorubicin extracellular

concentration starts to fall, which in turn causes the sink term to decrease and finally reaching

zero concentration.

Similar patterns can be found in normal tissues as shown in Figure 6-15. Quantitative

comparison shows that encapsulated doxorubicin concentration in normal tissues is 3 orders of

magnitude lower than that in tumour, indicating that TSL-mediated drug delivery could

effectively avoid liposome accumulation in normal tissue.


126

0 5 10 15 20 25 30 35 40 45 50 55 60

0.0

1.0x10-8

2.0x10-8

3.0x10-8

4.0x10-8

5.0x10-8

6.0x10-8

7.0x10-8

No-Heating

10 min Heating

30 min Heating

60 min Heating

Enca

psu

late

d D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (min)

Figure 6-15. Spatial-mean concentration of encapsulated doxorubicin in extracellular space of normal tissue as a

function of time.

The spatial concentration of encapsulated doxorubicin in extracellular space at different time

points is presented in Figure 6-16. Since blood vessel walls in tumour are leakier with larger

pores than those in normal tissue [110, 150], TSL particles are designed in a size range that

allows them to pass through the vessel walls in tumour only, thereby achieving effective

accumulation in the extracellular space. As shown in Figure 6-16, encapsulated doxorubicin

mainly accumulates in tumour, and drugs in normal tissue (shown in Figure 6-15) come from

tumour by migration in the extracellular space.

1.03×10-4

8.83×10-5

7.36×10-5

5.88×10-5

4.42×10-5

2.95×10-5

1.48×10-5

5.00×10-8

C (kg/m3)

1 min 2 min 5 min

1.03×10-4

8.83×10-5

7.36×10-5

5.88×10-5

4.42×10-5

2.95×10-5

1.48×10-5

5.00×10-8

C (kg/m3)

10 min 15 min 30 min

Figure 6-16. Spatial distribution of encapsulated doxorubicin extracellular concentration in tumour and surrounding

normal tissues at various time points with 60 minutes heating.


127

Encapsulated doxorubicin concentration in the tumour is highly heterogeneous upon HIFU

heating. Tumour centre has lower concentration than in periphery with time proceeds, owing to

the fast release of doxorubicin enhanced by HIFU heating in and around the focus regions.

Because of the reduction of encapsulated doxorubicin concentration in the blood stream, the

corresponding extracellular concentration decreases in the entire tumour and finally becomes to

uniform at 30 minutes when the temperature falls back to 37 oC.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

4.0x10-3

No-Heating

10 min Heating

30 min Heating

60 min Heating

Fre

e D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

2.6 2.7 2.8 2.9 3.02.4x10

-5

2.8x10-5

3.2x10-5

3.6x10-5

4.0x10-5

Fre

e D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

2.0x10-3

4.0x10-3

6.0x10-3

8.0x10-3

1.0x10-2

1.2x10-2

No-Heating

10 min Heating

30 min Heating

60 min HeatingB

ound D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

(b)

2.5 2.6 2.7 2.8 2.9 3.07.0x10

-5

8.0x10-5

9.0x10-5

1.0x10-4

1.1x10-4

1.2x10-4

1.3x10-4

Bound D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

No-Heating

10 min Heating

30 min Heating

60 min Heating

Fre

e D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(c)

3.5 3.6 3.7 3.8 3.9 4.02.0x10

-6

2.5x10-6

3.0x10-6

3.5x10-6

4.0x10-6

4.5x10-6

5.0x10-6

5.5x10-6

Fre

e D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

2.0x10-3

4.0x10-3

6.0x10-3

8.0x10-3

1.0x10-2

1.2x10-2

No-Heating

10 min Heating

30 min Heating

60 min Heating

Bound D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

3.5 3.6 3.7 3.8 3.9 4.06.0x10

-6

9.0x10-6

1.2x10-5

1.5x10-5

1.8x10-5

Bound D

oxoru

bic

in I

ntr

avas

cula

r C

once

ntr

atio

n

in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(d)

Figure 6-17. Doxorubicin extracellular concentration with various heating duration as a function of time. (a) free and

(b) doxorubicin concentration in tumour, and (c) free and (d) doxorubicin concentration in normal tissue.

The time courses of free and bound doxorubicin intravascular concentration in tumour and its

surrounding normal tissue are shown in Figure 6-17. Doxorubicin is released from TSL upon

heating, leading to an increase in intravascular concentration of drug in free and bound forms.

However, drug concentrations fall after reaching their peaks owing to the reduction of

encapsulated doxorubicin in blood. The predicted results show that hyperthermia is capable of


128

enhancing drug release and thereby increasing the concentration in blood in the tumour region,

whereas heating makes little difference to intravascular concentration in normal tissue.

0 1 2 3 4 5 6

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

No-Heating

10 min Heating

30 min Heating

60 min Heating

Fre

e D

oxoru

bic

in E

xtr

acel

lula

r C

once

ntr

atio

n

in T

um

our

(kg/m

3)

Time (hr)

5.5 5.6 5.7 5.8 5.9 6.0

6.0x10-6

9.0x10-6

1.2x10-5

1.5x10-5

1.8x10-5

2.1x10-5

2.4x10-5

Fre

e D

oxoru

bic

in E

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once

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in T

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our

(kg/m

3)

Time (hr)

(a)

0 1 2 3 4 5 6

0.0

2.0x10-3

4.0x10-3

6.0x10-3

8.0x10-3

1.0x10-2

No-Heating

10 min Heating

30 min Heating

60 min Heating

Bound D

oxoru

bic

in E

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once

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orm

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g/m

3)

Time (hr)

5.5 5.6 5.7 5.8 5.9 6.02.0x10

-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

7.0x10-5

Bound D

oxoru

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in E

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once

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g/m

3)

Time (hr)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

4.0x10-4

8.0x10-4

1.2x10-3

1.6x10-3

2.0x10-3

No-Heating

10 min Heating

30 min Heating

60 min Heating

Fre

e D

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once

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g/m

3)

Time (hr)

3.5 3.6 3.7 3.8 3.9 4.06.0x10

-6

9.0x10-6

1.2x10-5

1.5x10-5

1.8x10-5

2.1x10-5

Fre

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g/m

3)

Time (hr)

(c)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

1.0x10-3

2.0x10-3

3.0x10-3

4.0x10-3

5.0x10-3

6.0x10-3

No-Heating

10 min Heating

30 min Heating

60 min Heating

Bound D

oxoru

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g/m

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Time (hr)

3.5 3.6 3.7 3.8 3.9 4.02.0x10

-5

3.0x10-5

4.0x10-5

5.0x10-5

6.0x10-5

Bound D

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once

ntr

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in N

orm

al T

issu

e (k

g/m

3)

Time (hr)

(d)

Figure 6-18. Doxorubicin extracellular concentration with various heating duration as a function of time. (a) free and

(b) doxorubicin concentration in tumour, and (c) free and (d) doxorubicin concentration in normal tissue

Temporal profiles of free and bound doxorubicin concentration in tumour extracellular space and

its surrounding normal tissue are compared in Figure 6-18 between HIFU heating and no-heating.

Regardless of the heating duration, both free and bound doxorubicin concentrations increase

rapidly during the initial period following drug administration. Doxorubicin concentrations in

tumour interstitial space increase faster and reach a much higher peak upon heating. Although

doxorubicin concentration drops to a low level as time proceeds, longer heating duration is able

to maintain a slightly higher concentration. Comparing doxorubicin extracellular concentrations

in Figures 6-18 with the concentration in blood in Figure 6-17, the concentration curves have


129

identical shapes for a given heating duration. This means that drug concentration in blood has a

direct influence on the extracellular concentration for both free and bound doxorubicin.

Comparisons show that doxorubicin concentration in both free and bound forms in tumour is

increased by up to 11.26 % upon heating, owing to enhanced drug release, whereas the influence

of HIFU heating on doxorubicin concentration in normal tissue is trivial. On the one hand,

encapsulated doxorubicin concentration in extracellular space of normal tissue is limited (shown

in Figure 6-15 and 6-16), hence less free doxorubicin that can be released. On the other hand,

although there is a large amount of encapsulated doxorubicin in blood, drug release in most

regions of normal tissue is not enhanced as the temperature is close to 37oC.

Spatial distributions of doxorubicin intratumoural concentration and survival fraction of tumour

cells at 10 minutes after administration are displayed in Figure 6-19. It is clear that doxorubicin

concentration and survival tumour cells are non-uniform, which is not surprising since the

tumour and drug transport properties are enhanced by temperature rise, and as such, the position

of HIFU focus regions is expected to have a strong influence on the spatial distribution of drug

and cell killing.

3.39×10-3

3.36×10-3

3.34×10-3

3.31×10-3

3.29×10-3

3.26×10-3

C (kg/m3)

3.20×10-3

3.15×10-3

3.10×10-3

3.05×10-3

3.00×10-3

2.95×10-3

C (kg/m3)

2.68

2.49

2.30

2.12

1.93

1.75

C (ng/105cells)

Free Doxorubicin Intravascular

Concentration

Free Doxorubicin Extracellular

Concentration

Intracellular Concentration in

Tumour

1.02×10-2

1.01×10-2

1.00×10-2

9.93×10-3

9.84×10-3

9.76×10-3

C (kg/m3)

9.60×10-3

9.44×10-3

9.28×10-3

9.12×10-3

8.96×10-3

8.80×10-3

C (kg/m3)

99.43

99.41

99.38

99.36

99.34

99.32

f (%)

Bound Doxorubicin Intravascular

Concentration

Bound Doxorubicin Extracellular

Concentration

Survival Fraction of

Tumour Cells

Figure 6-19. Spatial distribution of doxorubicin concentration and cell survival fraction in tumour at 10 minutes.


130

Intracellular concentration is the highest in tumour centre where HIFU focus regions are located,

and it decreases radially towards tumour periphery with the lowest level at the right and bottom

corners. As a consequence, cell killing is most effective in the tumour centre, while the cytotoxic

effect is insufficient at the right and bottom corners which are the furthest from the HIFU focus

regions. Therefore, the heterogeneous enhancement presented in Figure 6-19 demonstrate that

TSL-mediated drug delivery with HIFU heating is capable of improving treatment efficacy in

designated tumour regions, and careful selection of focus regions may help achieve desired

therapeutic effect in the entire tumour.

Temporal profiles of intracellular concentration with various heating durations are presented in

Figure 6-20. The intracellular concentration displays a sharp rise at the beginning until it reaches

a peak, and then decreases. Compared with the case with no-heating, the rate of increase in

intracellular concentration improves upon heating. Increasing heating duration results in higher

peaks and higher intracellular concentration levels over time.

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

6

7

No-Heating

10 min Heating

30 min Heating

60 min Heating

Intr

acel

lula

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once

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n i

n T

um

or

(ng/1

05ce

lls)

Time (hr)

8.5 8.6 8.7 8.8 8.9 9.04.0x10

-3

6.0x10-3

8.0x10-3

1.0x10-2

1.2x10-2

Intr

acel

lula

r C

once

ntr

atio

n i

n T

um

or

(ng/1

05ce

lls)

Time (hr)

Figure 6-20. Doxorubicin intracellular concentration with heating and no-heating as a function of time.


The anticancer effectiveness is evaluated by calculating changes in the percentage of tumour

cells using a pharmacodynamics model detailed in Chapter 3. As can be seen from Figure 6-21,

the percentage of tumour cells starts to fall immediately after drug administration. There is a

rapid cell killing phase in the first few hours after heating in all treatments, but the duration of

this phase depends strongly on the heating duration. This is followed by a slower cell killing

phase which lasts for approximately 2 hours (6~8 hours after administration). With further


131

reduction in the intracellular concentration, the rate of cell killing and physiological degradation

is overtaken by that of cell proliferation, and as a result, the percentage of tumour cells begins to

increase. Results in Figure 6-20 suggest that anticancer effectiveness can be improved by

modifying the heating duration.

0 2 4 6 8 10 12 14 16 18

70

75

80

85

90

95

100

No-Heating

10 min Heating

30 min Heating

60 min HeatingS

urv

ival

Fra

ctio

n o

f T

um

our

Cel

ls (

%)

Time (hr)

Figure 6-21. Survival cell fraction in treatment with various heating duration as a function of time.

Relative to the control case with no-heating, improvement in the intracellular concentration and

cell killing with various heating durations are compared in Figure 6-22 (a) and (b), respectively.

Results show that improvements in intracellular concentration and cell killing are non-linearly

related to heating duration. This may be attributed to the fast reduction of encapsulated

doxorubicin concentration in blood and the non-linear relationship of drug transport described in

Chapter 3.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

60 min30 min

Impro

vem

ent

in I

ntr

acel

lula

r C

once

ntr

atio

n

(ng/1

05ce

lls)

10 min

(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

60 min30 min

Impro

vem

ent

in C

ell

Kil

ling (

%)

10 min

(b)

Figure 6-22. Improvement by HIFU heating. (a) intracellular concentration, and (b) cell killing


132

Although simulation results presented so far demonstrate that TSL drug delivery with HIFU

introduced hyperthermia can improve the cytotoxic effect of doxorubicin, the gain in quantitative

terms is quite small when compared to the control case. Possible reasons are given below.

(1) Unstable liposome. The simulated TSL formulation [143] is unstable at body temperature.

Ideally, TSL should be designed to release its contents upon hyperthermia only, with no release

at physiological temperature. However, this is difficult to achieve in vivo. Undesirable release at

37 oC can reduce the intravascular concentration of encapsulated doxorubicin and thereby

resulting in high concentration of free doxorubicin in blood. As a consequence, less doxorubicin

can be released upon hyperthermia. High concentration of doxorubicin in blood and normal

tissue may increase the risk of side effects, such as cardiotoxicity. Therefore, further

improvement in liposome formulation to prevent drug release at body temperature is essential to

future clinical application of TSL-mediated chemotherapy.

(2) Non-optimal focus region. Tumour cell killing is mostly enhanced in HIFU focus regions.

The rate of doxorubicin release is insufficient in the rest of the tumour as the local temperature is

lower than 42 oC, at which drug releases is most effective. Therefore, heating strategy needs to

be optimised in order to generate a uniform temperature profile close to the desired temperature

within tumour; these include careful design of the number and location of focus regions, heating

duration and HIFU scanning method.

6.3. Comparison of 2-D and 3-D Models

All the numerical results on TSL-mediated drug delivery presented so far are based on a 2-D

model reconstructed from a selected MR image of a transverse section. However, a real tumour

and its surrounding normal tissues are 3-D, and the acoustic pressure as well as the resultant

temperature field are highly heterogeneous upon HIFU heating. Hence, it is necessary to evaluate

the need for 3-D simulation by quantitative comparison of results based on 2-D and 3-D models.

For this purpose, the 3-D model shown in Figure 4-23 is adopted with an improved mesh to meet

the requirement of simulation on acoustic profiles upon HIFU heating, and the final mesh

consists of 4,161,964 tetrahedral elements which have been tested to produce grid independent

solutions.


133

Numerical simulation has been carried out with only one focus region located in the tumour

centre. Comparison of the maximum temperature within the tumour is shown in Figure 6-23. No

difference can be observed between the 2-D and 3-D models owing to the temperature control

mode described in Equation (6-1).

0 1 2 3 4 5 6 7 8 9 10

37

38

39

40

41

42

43

Tem

per

ature

(oC

)

Time (s)

2-D Model

3-D Model

Figure 6-23. Comparison of maximum temperature of tumour in 2-D and 3-D model.

Temporal profiles of spatial-mean temperature of tissues and blood based on the 2-D and 3-D

models are compared in Figure 6-24 and Figure 6-25, respectively. The predicted temperature is

higher in 2-D model. This over-prediction is attributed to the acoustic pressure distribution which

is highly non-uniform. The focus region to tumour size ratio is 0.17 (mm2/mm2) in the 2-D and

0.03 (mm3/mm3) in 3-D models. Since the focus region covers more tumour regions in the 2-D

model, the predicted temperature rises faster than in the 3-D model. Therefore, the predicted

drug release and cell killing are expected to be more effective in 2-D model simulations.

0 1 2 3 4 5 6 7 8 9 10

37.0

37.2

37.4

37.6

37.8

38.0

Tem

per

ature

(oC

)

Time (s)

2-D Model

3-D Model

(a)

0 1 2 3 4 5 6 7 8 9 10

37.00

37.01

37.02

37.03

37.04

(b)

Tem

per

ature

(o

C)

Time (s)

2-D Model

3-D Model

Figure 6-24. Comparison of spatial mean temperature in 2-D and 3-D model. (a) tumour and (b) normal temperature.


134

0 1 2 3 4 5 6 7 8 9 10

37.00

37.05

37.10

37.15

37.20

(a)

Tem

per

atur

(oC

)

Time (s)

2-D Model

3-D Model

0 1 2 3 4 5 6 7 8 9 10

37.000

37.002

37.004

37.006

37.008

37.010

Tem

per

ature

(oC

)

Time (s)

2-D Model

3-D Model

(b)

Figure 6-25. Comparison of spatial mean temperature in 2-D and 3-D model. (a) blood temperature in tumour, and

(b) blood temperature in normal tissue.

By incorporating bioheat transfer and drug transport in a realistic tumour model, the presents

study has provided insights into TSL-mediated drug delivery to solid tumours. Nevertheless, the

mathematical model employed here involves a number of assumptions.

First, physical complexities including cavitation and the non-linear progression of ultrasound are

ignored. It has been shown that the effects of these complexities are minor for a local intensity in

the range of 100~1000 W/cm2 and the peak negative pressure in the range of 1~4 MPa [151,

152]. Given that the simulated transducer parameters (acoustic intensity is 120 W/cm2, peak

negative pressure is 1.98 MPa) are within the specified ranges, these physical complexities are

not considered to be important at the conditions examined here.

Second, the influence of temperature elevation on tumour properties is complicated. Hence not

all variations, such as temperature-dependent change in plasma clearance, are included in the

current model. Detailed descriptions and mathematical models are required in the future studies

in order to examine these effects.

Third, focus regions are ideally positioned in the tumour, without considering movement

trajectory of ultrasound transducer, vascular distribution and other micro-environment in tumour.

Moreover, only 3 heating durations have been studied. In order to have a more comprehensive

investigation, complex heating strategies should be simulated. These include release mode,

duration/timing of heating and various release rates.


135

6.4. Summary

In this chapter, thermo-sensitive liposome-mediated drug delivery with various heating durations

are compared based on a 2-D prostate tumour model with a realistic geometry. The results

indicate that thermo-sensitive liposome-mediated drug delivery is a possible method to achieve

localised treatment in cancer therapy. Heating duration is a controllable parameter to improve the

efficacy of anticancer treatment.

7. Conclusion

In this chapter, main conclusions and limitations of the present study are summarized.

This is followed by suggestions for further works in anticancer drug delivery

research.

7. CONCLUSION

137

7.1. Main Conclusion

In this thesis, a mathematical model comprising drug transport and bioheat transfer is presented

to simulate the delivery of anticancer drugs, either in their free form or encapsulated in thermo-

sensitive liposomes. Anticancer efficacy of the delivered drugs is predicted based on a

pharmacodynamics model. Model parameters describing physiological and biological properties

of tissue and drugs are derived from experimental data in the literature. The model has been

applied to realistic tumours reconstructed from MRI in order to (a) understand the transport steps

of non-encapsulated drugs (Chapter 4), (b) elucidate the effects of tumour properties (Chapter 5),

and (c) examine the coupled effect of heat transfer and drug transports under thermo-sensitive

liposome-mediated delivery (Chapter 6).

The main findings are as follows:

(1) Better anticancer efficacy is achieved with administration methods that are able to maintain a

high level of extracellular concentration over the entire treatment period.

In direct delivery of non-encapsulated drugs, continuous infusion is much more effective than

bolus injection in improving the cytotoxic effect of doxorubicin on tumour cells and reducing

peak levels of doxorubicin in blood stream. Although rapid continuous infusion with a high

peak intracellular concentration leads to faster cell killing at the beginning of the treatment,

slow infusion modes give better overall treatment efficacy.

Increasing drug dose can increase concentration levels in the entire tumour and thereby

improving doxorubicin cytotoxicity. Whilst consideration must be given to the limitation on

doses used in clinical treatment owing to potential side effects, which are caused by

doxorubicin in the circulatory system and normal tissues, the highest tolerable dose gives the

best therapeutic efficacy.

Multiple-administration can improve the efficacy of anticancer treatments, while significantly

reducing the peak intravascular concentration, thereby lowering the risk of cardiotoxicity

caused by doxorubicin in blood stream. Treatment outcomes may be improved by increasing

the number of administrations. Although there is a slight reduction in predicted anticancer

7. CONCLUSION

138

effectiveness, reducing the total dose while increasing the number of continuous infusions is

effective in reducing drug concentration in blood.

(2) Interstitial fluid pressure is not shown to be sensitive to microvasculature density in the

interior of tumour, implying that drug transport by pressure-induced convection is weak

except in a thin layer close to the interface of tumour/normal tissue. However, better

anticancer effectiveness is achieved in tumour regions with relatively low but non-zero

microvasculature density, owing to reduced loss of drugs back to the circulatory system.

(3) Spatial-mean transvascular transport is strong in small tumour which is well-vascularised,

owing to the low spatial-mean interstitial fluid pressure and dense microvasculature.

However, anticancer effectiveness is compromised in small tumour as a result of enhanced

reverse diffusion of drug to the blood circulation.

(4) Thermo-sensitive liposome-mediated drug delivery with hyperthermia induced by high

intensity focused ultrasound is effective in releasing encapsulated anticancer drugs within

targeted regions, thereby achieving localised treatment. Heating duration is an important

factor in determining the efficacy of anticancer treatment. Increasing heating duration to 1

hours is beneficial in help maintain higher concentration of anticancer drugs and improve

predicted treatment efficacy.

(5) Under the assumption of homogeneous and uniform tissue and drug transport properties, 2-D

models of tumours without explicit representation of microvasculature provide comparable

results to corresponding 3-D models as far as spatial-averaged parameters are concerned.

Nevertheless, 3-D model is preferred for comprehensive details about spatial distributions of

drug concentration and cell killing, especially when heterogeneous microvasculature density

and heat transfer upon HIFU heating are considered.

7.2. Limitations

The mathematical models described in Chapter 3 involve a number of assumptions which relate

to descriptions of tumour properties, drug transport and HIFU heating. Details of these are given

below.

7. CONCLUSION

139

7.2.1. Tumour Property

In the present study, tumour properties except for cell density are considered to be time-

independent. In reality, tumour growth is a dynamic process which requires variations of tumour

properties with time. The effect of this assumption is considered to be minor for simulations with

a time window of a few hours only. However, variations of tumour properties should be

considered when simulating treatment over a very long time period.

Except for the tumour microvasculature density studied in Chapter 5, tumour properties are

assumed to be homogeneous. However, these properties are likely to be heterogeneous and non-

uniform, which will affect the transport of anticancer drugs to tumour cells, and eventually the

effectiveness of an anticancer treatment.

The present study can be regarded as a qualitative investigation for predictions of trends since all

model parameters, apart from tumour geometry, correspond to average and representative values

extracted from the literature. More information is required for type-specific and patient-specific

studies.

7.2.2. Drug Transport Model

Tumour vascular network is treated as a source term in the governing equations instead of being

modelled explicitly. As a consequence, the effect of intravascular transport and geometric

features of tumour vasculature are not included. While this assumption may be an over-

simplification, it is practical given the lack of information on realistic tumour vasculature

geometry and the complexity in direct modelling of blood flow in a vascular network coupled

with interstitial drug transport.

Furthermore, only the effect of intratumoural microenvironment on drug transport is considered.

However, anticancer drugs may in turn alter the intratumoural microenvironment. For example,

drugs such as doxorubicin also attack blood vessels [138] by reducing microvasculature density

in tumour. This effect can be incorporated by using a more detailed description with a two-way

interaction.

Drug efflux from cells depends on the transmembrane protein, P-Glycoprotein (P-gp), which is

capable of effectively reducing intracellular concentration of structurally unrelated cytotoxic

7. CONCLUSION

140

agents [153], such as doxorubicin [154] and cisplatin [155]. In the present study, this efflux is

included without considering variations of P-gp concentration over time [99]. This limitation

can be overcome by employing an improved transmembrane model.

7.2.3. HIFU Heating Model

The proposed HIFU heating model is a first step towards developing a comprehensive numerical

platform for thermo-sensitive liposome-mediated drug delivery system. The employed transducer

only consists of one element. However, transducers with a few hundreds of elements are required

and have been adopted in clinical use [156]. Moreover, sonication is controlled by a binary

feedback loop with a modified algorithm [157] based on the adjustment of the ultrasound phase

generated by each element.

7.3. Suggestions for Future Work

The long term goal of computational investigations of drug delivery to solid tumour is to develop

an experiment-assisted computational modelling framework spanning across multiple spatial and

time-scales to advance the understanding of drug transport process and provide guidance for

improvement in anticancer treatments. Therefore, several possible modelling tasks have been

identified and these are described below to give an overall idea about how to expand the

modelling work described in this thesis.

7.3.1. Realistic Blood Vessel Geometry

The morphological feature of blood vessels in tumour is significantly different from those in

normal tissue, leading to highly abnormal organization and connection between vessels [20, 21].

Experimental [158] and numerical [102] studies showed that not all the tumour cells could be

exposed to sufficient drugs during treatment. Real micro-vasculature networks reconstructed

from advanced imaging techniques may help to examine the effect of blood vessel geometric

features on drug delivery and the resultant cytotoxic effect. Modelling of blood flow can then be

coupled with intravascular drug transport. Such a model can provide spatial-temporal profiles of

drug concentration and cytotoxic effect to gain more insights into the effect of tumour-specific

blood vessel network on anticancer effectiveness in different types of tumours.

7. CONCLUSION

141

7.3.2. Nutrient Transport

Tumour growth needs nutrient which is also carried by the circulating blood. Highly proliferating

tumour cells and insufficient supply of nutrient, such as oxygen and glucose, may result in

hypoxia in localised tumour regions, and cause tumour cell arrest. Given that many anticancer

drugs (e.g. doxorubicin) act on proliferating cells because their working mechanism is to stop

cell duplication by interacting with DNA [3], treatment efficacy may be compromised in regions

with insufficient supply of oxygen and other nutrients. On the other hand, arrested tumour cells

can re-proliferate after nutrition in the surrounding environment is restored, or tumour cells may

die from nutrition shortage, resulting in the formation of necrotic core. It is important to point out

that blood supply is closely related to the distribution of functional microvasculature. Therefore,

it would make the model more realistic by incorporating nutrient transport, especially when

heterogeneous distribution of microvasculature is considered.

7.3.3. Model Validation

The present study serves to provide predictions of qualitative trends, hence detailed model

validations are not included except for the HIFU acoustic model. This limitation is attributed to

both the numerical model and experiments. On the one hand, not all model parameters are

available for a specified tumour, especially when considering the unique condition of each

patient; on the other hand, important results, such as intracellular concentration and cell density,

cannot be directly measured in vivo. Animal experiments may help to provide comparable data

for validation. Advanced optical methods, such as high-performance liquid chromatography, can

potentially be used to obtain in vivo data for validations in the future.

7.3.4. Type / Patient-Specific Tumour

The properties of tumour and normal tissue vary with tissue type, growth stage and individual

conditions of a patient [41, 45]. These variations may influence drug transport processes and

eventually the anticancer efficacy. Numerical study can be used to predict these differences and

provide suggestions in order to optimize treatments. However, detailed type-specific and patient-

specific information is required.

7. CONCLUSION

142

7.3.5. Chemotherapy Drugs

Since different anticancer drugs can be combined in clinical use [3, 159, 160], it would be useful

to simulate drug transport and cell killing for different combinations of drugs. Most of

chemotherapy drugs experience similar transport processes in delivery to tumour cells, including:

transport within/through blood vessels, migration in interstitial space and cell uptake. Besides

doxorubicin, the basic mathematical model presented here can be applied to various drugs, such

as 5-fluorouracid, BCNU, cisplatin, Methotrexate and antibody (e.g. Fab and IgG). Necessary

modifications should be made in terms of drug-specific properties and behaviour. For instance,

5-fluorouracid and cisplatin do not easily bind with proteins [17].

7.3.6. HIFU Heating Strategy

Heating strategy is essential in thermo-sensitive liposome-mediated drug delivery systems.

Further computational studies should be carried out to design an optimal heating strategy that

aims to generate a uniform temperature distribution within designated tumour regions in order to

improve local treatment efficacy while reducing side effects to normal organs. HIFU scanning

method of multiple-element transducer can be designed to maintain tumour temperature above

the phase transition temperature of liposome while minimising heating of untargeted regions.

Moreover, the predicted anticancer efficacy can be influenced by a number of factors which

should be optimised for the treatment of different types of tumours; these include release mode

(intra- and extra-vascular triggered release), duration and timing of heating and release rate of

chemotherapy drugs.

In summary, a mathematical model for drug delivery in free and encapsulated forms has been

developed in order to provide qualitative and mechanistic understanding of anticancer drug

transport and cytotoxicity in solid tumours. The developed numerical platform can serve as a

foundation for further comprehensive investigations on anticancer drug delivery.

143

Appendix: Publications during the Project

Journal papers

[1] Wenbo Zhan, Xiao Yun Xu, 2013, A mathematical model for thermosensitive liposomal

delivery of Doxorubicin to solid tumour., J Drug Deliv, Vol:2013, ISSN:2090-3014

[2] Wenbo Zhan, Wladyslaw Gedroyc, Xiao Yun Xu, 2014, Mathematical Modelling of Drug

Transport and Uptake in a Realistic Model of Solid Tumour. Protein Pept Lett, Vol: 21, Issue

No.11, 2014: 1146-56.

[3] Wenbo Zhan, Wladyslaw Gedroyc, Xiao Yun Xu, 2014. Effect of Heterogeneous

Microvasculature Distribution on Drug Delivery to Solid Tumour. Journal of Physics: D. 47,

2014: 475401

Conference Presentations

[1] Wenbo Zhan, Wladyslaw Gedroyc, Xiao Yun Xu. Computational Study of Drug Transport in

Realistic Models of Solid Tumour. AIChE Annual Conference, Pittsburgh, US, 2012

[2] Wenbo Zhan, Wladyslaw Gedroyc, Xiao Yun Xu. Computational Study of Drug Transport in

Realistic Models of Solid Tumour. Mathways into Cancer II, Carmona, Sevilla, Spain, 2013

[3] Wenbo Zhan, Xiao Yun Xu. Simulation of HIFU Heating in Solid Tumour: Comparison of

Different Temperature Control Modes. 3rd

International Conference on Computational

Methods for Thermal Problems, Lake Bled, Slovenia, 2014

144

Appendix: Nomenclature

Symbols

A1 parameter of pharmacokinetic model for doxorubicin



Af frequency factor

b chord from the ultrasound transducer centre to the radiator boundary

c specific heat

Cbe bound doxorubicin concentration in interstitial fluid

Cbp bound doxorubicin concentration in blood plasma

Cfe Free doxorubicin concentration in the interstitial fluid

Cfp free doxorubicin concentration in blood plasma

CGd tracer concentration in the interstitial fluid

Ci intracellular doxorubicin concentration

CLbp plasma clearance of bound doxorubicin

Cle liposome encapsulated drug concentration in the interstitial fluid

CLfp plasma clearance of free doxorubicin

Cp_Gd tracer concentration in blood

Cv doxorubicin concentration in blood plasma

d distance from a point on the radiator to a field point

Dbe diffusion coefficient of the bound doxorubicin-protein

Dc tumour cell density

Dd drug dose

Dfe diffusion coefficient of free doxorubicin

Dl diffusion coefficient of liposomes

DS degree of vascular stasis

DW diffusion coefficient in water

EC50 drug concentration producing 50% of fmax

Fbe bound doxorubic gain from blood vessels

145

Ffl doxorubicin loss of doxorubicin to the lymphatic vessels per unit volume of tissue

Ffp doxorubicin gained from the blood capillaries in tumour and normal tissues

Ffp free doxorubicin gained from the capillaries in tumour and normal tissues

Fll loss of liposomes through the lymphatic vessels per unit volume of tissue

Flp liposomes gained from the capillaries in tumour and normal tissues

Fly interstitial fluid absorption rate by the lymphatics per unit volume of tumour tissue

fmax cell-kill rate constant

Fv interstitial fluid loss from blood vessels per unit volume of tumour tissue

h depth of the concave surface

Ia ultraound intensity

k thermal conductivity

ka doxorubicin-protein binding rate

kac acoustic velocity potential

kB Boltzmann's constant

kci PI control parameter

kcp PI control parameter

kd doxorubicin-protein dissociation rate

ke michaelis constant for transmembrane transport

kg cell physiological degradation rate

ki michaelis constant for transmembrane transport

Kly hydraulic conductivity of the lymphatic wall

kp cell proliferation rate constant

krel release rate of doxorubicin from liposome

Kv hydraulic conductivity of the microvascular wall

md inceased fold of diffusivity

mp inceased fold of permeability

mtv inceased fold of transmembrane rate

MW molecular weight

P permeability

pa acoustic pressure

Pbe permeability of vessel wall to bound doxorubicin

146

Pef trans-capillary Peclet number

Pfe permeability of vasculature wall to free doxorubicin

pi interstitial fluid pressures

ply intra-lymphatic pressure

pv vascular pressures

q ultrasound power deposition

r radius of spherical particle

R universal gas constant

s percentage of bound doxorubicin

S/V surface area of blood vessels per unit volume of tumour tissue

(S/V)* relative value of surface area of blood vessel per tumour size

Sb net rate of doxorubicin gained from association/dissociation with bound doxorubicin-

protein

SGd MRI signal with tracer

SGd_0 MRI signal without tracer

Si net rate of doxorubicin gained from the surrounding environment

Sl net rate of liposomes gained from the surrounding environment

Slp liposomes gained from plasma

Sly/V surface area of lymphatic vessels per unit volume of tumour tissue

Sr released drug in the interstitial fluid

Su influx/efflux from tumour cells

Sv net rate of doxorubicin gained from the blood/lymphatic vessels

T temperature

T* absolute temperature

T0 body temperature

t time

td infusion duration

Tn normal body temperature

u0 velocity constant

va speed of ultrasound

VB plasma volume

147

Vmax rate of trans-membrane transport

VT tumour volume

W diagonal matrix

w perfusion rate of blood flow

w0 perfusion rate of blood flow at body temperature

ΔE activation energy

Tensor

C matrix of the inertial loss term

F Darcian resistance to fluid flow through porous media

I unit tensor

u normal velocity of a slightly curved radiating surface

v velocity of the interstitial fluid

Greek Letter

α ultrasound absorption coefficient

α1 compartment clearance rate of doxorubicin



ɛ cellular efflux function

ζ cellular uptake function

κ permeability of the interstitial space

λ wave length of ultrasound

μ dynamic viscosity of interstitial fluid

πi osmotic pressure of interstitial fluid

πv osmotic pressure of the plasma

ρ density

σ osmotic reflection coefficient

σd osmotic reflection coefficient for the drug molecules

148

σT average osmotic reflection coefficient for plasma protein

τ stress tensor

φ tumour fraction extracellular space

Ψ acoustic velocity potential

ω angular frequency of ultrasound transducer

Abbreviation

BSA body surface area

BSNC bis-chloroethylnitrosourea

CFD computational fluid dynamics

FVM finite volume method

Gd-DTPA gadopentetate dimeglumine

HIFU high intensity focused ultrasound

IFP interstitial fluid pressure

IFV interstitial fluid velocity

MRI magnetic resonance image

MW molecular weight

PEG polyethylene glycol

PDT photodynamic therapy

P-gp P-Glycoprotein

TSL thermo-sensitive liposome

UDS user defined scalar

149

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